Talk:Regular diatonic tuning

temperament or temperaments?
Help me out here. Is this article supposed to be about one tuning called "Syntonic temperament", or is it supposed to be about a system for generating a continuum of different tunings? It seems to be blurring those concepts. The title and opening statement suggest one particular tuning, but the illustration and the rest of the body text suggest multiple tunings. 108.60.216.202 (talk) 06:16, 25 May 2015 (UTC)
 * Hi, a distinction that's helpful but often blurred is that between a temperament and a tuning. A tuning is simply any assignment of notional pitches (like, say, "middle C") to frequencies (such as "260 Hz").  A temperament is a system of adjusting pitches away from their just intonation positions to achieve some desired feature(s).  For instance, you may know that different orchestras tune to slightly different frequencies.  In North America, the standard is to tune the A above middle C to 440 Hz.  In Europe, orchestras often tune to a slightly higher frequency, 444 Hz.  If a European and an American orchestra play the same piece, they will *tune* differently, but they will (almost certainly) perform it based on the same temperament, namely 12-tone equal temperament (the actual placement of notes will deviate slightly for various purposes, but the mental framework will be 12-equal, or at least compatible with 12-equal).
 * 12-tone equal temperament is a *temperament* because all it requires is that an octave be divided into twelve equal steps. Where those steps lie in terms of absolute frequency is a separate issue from the temperament.
 * So, a temperament need not assign every pitch in the system to a specific frequency value. Instead, it may specify the mathematical relationship between *intervals*.  In a syntonic temperment, the interval of a major third is equated to a stack of four perfect fifths, minus two octaves.  Depending on the size of the fifths, the major third can vary in size fairly widely.  If the fifths are 700 cents (as in 12-tone equal temperament), then a major third is 400 cents.  If the fifths have their just values of approx. 702 cents, then a major third is 408 cents.  In the Renaissance and early Baroque, organs were constructed so that their fifths were (ideally) 696.6 cents, and so a major third came out to 386.3 cents, which is (within rounding error) just.  All of these are syntonic temperaments, because the relation between perfect fifths and major thirds is the same. Atemperman (talk) 00:15, 15 September 2015 (UTC)
 * I don't quite agree with this distinction between a temperament and a tuning. IMO, a tuning is any system used to tune anything, usually an instrument; a temperament is a particular tuning obtained by "tempering" an interval (in practice always the fifth, if I am not mistaken) in order to gain something else, e.g. better thirds. There is no reason to suppose that a tuning involves adjusting a pitch standard and that a temperament does not: this is another problem, solved in the same way in both tunings and temperaments. To say it otherwise, the matter of pitch standard arises as soon as the tuning or the temperament is used to actually tune an instrument; it is of little concern for the theory of these systems.
 * Another common terminological imprecision, in this article and in several others, is the idea that one "tempers out" an interval, typically a comma. In equal temperament, for instance, one does not "temper out" the Pythagorean comma, one tempers the fifths in such a way that the comma vanishes. A tempered interval is a tuned interval that cannot be expressed by a ratio of whole numbers: this is the technical difference between a temperament and a tuning. The Pythagorean tuning is a tuning, and not a temperament, because none of its intervals is tempered. It might appear that the "wolf" interval that it produces (the last 5th in the series, which actually is a diminished 6th, say G♯–E♭, or an augmented third, E♭–G♯) is an irrational interval, but that is not true: the ratio is complex, but it exists: it is equal to (3:2)11:(2:1)6, i. e. 177147:131072.
 * To answer the initial question, the "syntonic temperament" is defined here as having "the same definition as that of the meantone temperament", which is strange. A meantone temperament is one in which all fifths are tempered by the same amount, whith the result that the tone (two 5ths) is the mean of the major third (four 4ths): this is a family of temperaments that includes equal temperament, among other, where the fifths are tempered in order that the Pythagorean comma vanishes. There are meantone temperaments that are devised to make the syntonic comma vanish, such as the 1/3-comma meantone, where three tempered fifths result in a pure minor third (say, C–G–D–A), while pure fifths would have produced a minor third a comma too narrow; or the 1/4-comma meantone, where the same result obtains for the major third (say, C–G–D–A–E) which otherwise would be a comma too large. But the article writes
 * The syntonic temperament can be viewed as a systematic extension of the meantone temperament beyond the range imposed by the meantone temperament's need to remain consonant with timbres that conform to the harmonic series.
 * which, to me, sounds almost like a joke. I suppose "consonant", here, is not taken stricto sensu; but I utterly fail to see in what sense something called "timbre" could "conform to the harmonic series", not what that would mean.
 * Figure 1 appears to be a list of values for the tempered fifths (with the untempered one of 702 cents in the middle). Historical meantone temperaments so to say never widened the fifths (i.e. never used values of more than 702 cents), because in common practice music there is nothing to be gained. And they rarely tempered to more that 1/3 comma (695 cents), for the same reason; in my opinion, the values marked "pure 5/4 1/4 comma" and "pure 6/5 1/3 comma" are the only true syntonic temperaments, because they are the only ones really meant to make a syntonic comma vanish. Figure 1 delimits at the left a "Pure tuning range" that turns out to be the range of historical meantone temperaments, and at the rigth "Ordinal tuning ranges" which are nowhere defined here, but which are NOT syntonic temperaments, as other sources (including some of the papers quoted) would show.
 * The article refers to a "research program Musica Facta", which appears to be an anonymous blog that disappeared two years ago. This may have been a blog of User:JimPlamondon (see Jim Plamondon), who wrote the major part of this article and co-signed all the (three) references given: is this a case of WP:EOR? — Hucbald.SaintAmand (talk) 13:02, 11 January 2016 (UTC)

Meantone temperament
This page seems to be describing what is normally called a "Meantone temperament" To quote from the website

"
 * The syntonic temperament (which includes many notable “real world” tunings);


 * The schismatic temperament (which is closely related to the syntonic, and which has arguably been used in traditional Turkish music), "

But the chart shown includes tunings with the fifth wider than the pure fifth. So it's hard to be sure what it is about. The more general term here is a Linear temperament - in other words any rank 2 regular temperament with one of the generators the octave. In its most general case the other generator in a Linear temperament doesn't have to be a fifth, but can be of any size. So, I think there are two options here, really. We could just merge this with meantone, or we could merge it with Linear temperament which currently redirects to Regular temperament. As it stands, syntonic temperament seems to be a term coined by a single author and only used on the MusicaFacta website. Robert Walker (talk) 17:05, 12 July 2016 (UTC)
 * I, for one, fully agree with the above which, I think, expresses in much simpler terms my own lengthy comments above. This article certainly should be merged with menatone, or with regular temperament, or merely disappear. I had expected to receive at least some comment to mine above, but as none came, I sort of forgot the matter all together. But Robert Walker is perfectly right. The article as it is refers to no reliable source and appear merely to reflect the fantasy of one single author, the owner or the MusicaFacta website. I am not very much at ease with the procedures to merge an article with another, or to make it disappear, and I hope that someone may be kind enough to do so. But I do believe that it should be done without delay. — Hucbald.SaintAmand (talk) 21:21, 12 July 2016 (UTC)


 * Okay the process for a merge itself is quite simple. If there is anything here worth adding to the page you merge it with, then copy that over first. Then, just edit the page and replace the entire page with


 * 1) redirect Linear temperament
 * or


 * 1) redirect Meantone temperament


 * Or whatever choice one makes. I think both of those are fine.
 * If it is a non controversial edit we can just go ahead and do it. See WP:MERGEINIT Should also remove it from the template Template:Musical_tuning. If there is controversy you need to do a Merger proposal first as described on that page. Robert Walker (talk) 22:05, 12 July 2016 (UTC)


 * Just checked, there doesn't seem to be much distinctive about this page except the second figure half way down the page plus its explanation and sound files. But whatever it means, it is unsourced as far as I can tell. It seems likely to be an example of WP:OR, seems to me. If we did merge the page away, the whole page is still available for future reference in the history for its redirect statement, one advantage of this approach. Robert Walker (talk) 22:21, 12 July 2016 (UTC)


 * Just checked the author's contribution history and he is barely active here, only a few edits a year. It seems like it might possibly be an uncontroversial thing to do. Assuming good faith, it might well be that he did it at a time when he didn't understand wikipedia policies and thought original research was acceptable, and then just forgot about it. Robert Walker (talk) 22:30, 12 July 2016 (UTC)


 * Sorry for all these comments. It's a bit more complex than I thought, as the Meantone temperament article is written on the basis that meantone is a special case of Syntonic and links to this page as the more general case. So if we redirect this page to meantone, we have to edit that article also. The citations are to the "Computer Music Journal" which I think would normally count as a reasonable cite.


 * The idea seems to be just to be to extend meantone to the region between 5-et and 7-et - so to include schizmatic temperaments as well as meantone and to temper the fifths all the way until in one direction, the diatonic semitone vanishes, and in the other direction, the chromatic semitone vanishes, and that's the region he calls, for some reason, the "syntonic continuum", a subset of the linear temperaments. It is a natural region to mark out in its way, though it's rather strange to call it the "syntonic continuum" - it would be just as valid or invalid to call it the "Schismatic continuum" or indeed any other comma especially since it can be used both negatively as well as positively. I wonder if it has an alternative name, this concept? Also there are 37 results in the search for syntonic temperament in wikipedia. So it is a bit more than just merging one page away.


 * I just recently proposed a new project, the WikiProject Council/Proposals/Microtonal Music, Tuning, Temperaments and Scales, and if we don't find an immediate simple solution, perhaps this could be one of the tasks of the project to sort this all out and decide what to do. If you are interested, do join the project, BTW - we have 7 signatures so far and are aiming for 16 to make a viable number to start a new project. Robert Walker (talk) 00:21, 13 July 2016 (UTC)

Let me answer you here for the time being, before I make my objections on your WikiProject Council/Proposals/Microtonal Music, Tuning, Temperaments and Scales page, about which I have to think further. I am a historian of music theory, which means that I not only considered the history of temperaments, but also that of non European tunings e.g. in Oriental (maqam) music. My feeling is that microtonal music forms a very little part of that ensemble (it would be a gross mistake to consider maqam music as microtonal, even although the mistake is at times done by Arabic musicians). I believe that the confusion made here, calling "syntonic temperament" something that boils down to being (extended) meantone temperament is a case in point. Microtonal websites (such as those concerned with xenharmonic music) often appear to me as playgrounds for arithmetic games (which may be interesting if only because they turn back to the Antique Pythagorean conception of music as another name for arithmetics), but which have little to do with real music. As you can see, I am an old-fashioned theorist; but I keep up to it.

To turn back to the matter of merging or deleting this article, I had thought we should leave a chance to the original author(s) to react to the proposition. I suppose that after now about sixth months, they had their chance and ditn't take it. Then remains the problem that several other articles may need corrections as a result of the move. Some of my ideas (and a few of my mistakes) about all this can be read on the Talk:Schismatic_temperament page.

Hucbald.SaintAmand (talk) 09:31, 13 July 2016 (UTC)


 * Sorry, I don't understand. First on the xenharmonic wiki - it's a case of composers and theorists working together. A lot of the systems of music described there are used in compositions. As an example, here is Kraig Grady's "A Fairwell ring" for the Hexany. This is a modern tuning due to Erv Wilson which has eight just intonation triads in a scale of only six notes. What's more, all the diads of each triad are shared with other triads. It makes a very interesting musical structure to compose in which has a sense of weightlessness, like zero gravity - there's no preferred tonic, and every triad leads to more triads in all directions. There are other scales with more notes with even more consonances, with tetrads joined together and even larger chords in similarly maximal and intricate ways.


 * Then on the other side you have theorists like Gene Ward Smith especially. He works in an abstract area of maths. Anyway he developed various tools that let composers see interconnections between tuning systems and to devise new tuning systems and to see individual tunings as part of larger patterns that they hadn't been able to see before. This has been a vibrant synthesis of maths and music. Much of it developed in just the last ten years. I'm a mathematician by training myself but it's one of the things about maths that it is very common that you don't understand the maths developed even by people who are working in very closely related areas. It is just the nature of it because of the way it works with such abstract ideas. But that doesn't prevent it from being valuable. So some of it is maths that is of a lot of practical value, and some of it is pure theory which is there ready for applications if needed. It used to be that there was a lot of stumbling in the dark. Nowadays for many questions, say you want to optimize a tuning for some particular property- you can ask the theorists and they may be able to tell you the answer right away and save weeks of experimenting.


 * At the same time we have composers who are working with various other forms of microtonal music, such as the maqams indeed, and the medieval tuning systems, Gamelan music and so on.


 * I don't understand why you would say that maqam music is not microtonal. Surely microtonal music is the study of all forms of tuning world wide? Including also pygmie tunings, Thai music etc. We have wondered whether to give the group another name, so it might be that your objection is just to the name. We wondered wheher to call it Xenharmonic, but that's not a widely known name. And the problem with just leaving out the word Microtonal is thta it is too restrictive, because then it won't include for instance microtonal composers, or compositions, or the historical aspect etc etc. In the article on Microtonal music here it says "The term "microtonal music" usually refers to music containing very small intervals but can include any tuning that differs from Western twelve-tone equal temperament. Traditional Indian systems of 22 śruti; Indonesian gamelan music; Thai, Burmese, and African music, and music using just intonation, meantone temperament or other alternative tunings may be considered microtonal" - that's the sense in which we understand the term. I've added that to our project proposal description, which may help, since, as you say, it is sometimes used in a narrower sense.


 * With the Schismatic temperament article, the term is certainly in widespread use, I hear it all the time in microtonal conversations. I don't know what the historical origin is, it would be another good focus for the project. Experts may be able to provide more citations or help resolve the debate. The aim of the project is to encourage more people to wikipedia who are knowledgeable about such things and that can help resolve such issues, and make it easy to ask for help - as it is now you don't have any active project where you can ask for help about something like that specifically. Most people who are members of the Music or Classical music projects surely won't have a clue about such things. I'd certainly vote that it shouldn't be merged with Schizma. It's an important concept, tunings in which the fifth is wider rather than narrower than pure. For instance ETs are classified according to which the closest to a pure fifth is meantone like or schizmatic like, same also for linear temperaments. Anyway I think this is a discussion that could benefit from more participants from a very wide range of views. And that's the sort of thing the project would encourage.

Robert, you write:
 * Does that make sense? Robert Walker (talk) 11:47, 13 July 2016 (UTC)
 * At the same time we have composers who are working with various other forms of microtonal music, such as the maqams indeed, and the medieval tuning systems, Gamelan music and so on.

Well, we obviously do not share the same idea of what microtonal might be. To say that anything different from ET temperament is microtonal makes no sense to me because ET is at best an utopy. It has so to say never been used in real music, unless with electronic instruments. Even pianos have octaves that are slightly too large, because of the inharmonicity of stiff strings; guitars the same. Wind instruments hardly could play in true ET and stringed instruments usually adapt their intonation to the context.

One might consider that microtonal music is that which uses more than 12 tones in the octave. But that, of course, would exclude the musics that you quote, maqam, medieval or Gamelan. If more different notes are used in their heptatonc (or possibly pentatonic) scales, it merely results from the notes being mobile; and to consider a movable note as consisting in several different notes is a very dangerous stretch of the mind. To consider Pythagorean tuning a microtonal tuning seems to me counterproductive, because it merges to saying (with Julen Torma, "Euphorismes", in Écrits définitivement incomplets) that "Everything is the same thing. Everything therefore is very sufficiently good". To consider that the various tunings in which Bach's Well tempered keyboard has been played (or could be played) as variously microtonal is utterly problematic.

Any definition of "tuning" should take account of the fact that many musics make use of mobile (or movable) notes – that is, notes that are not tuned stricto sensu. We would easily agree, I trust, about what a temperament is: in addition to make use of tempered intervals, it is also meant for instruments of fixed tones, producing a limited number of tones. From the present discussion, it appears to me that a tuning too must concern limited numbers of tones. Violin strings are tuned (usually in pure or just fifths); but can one say that the notes played also are tuned (especially if the open strings are not actually played)? In what kind of tuning does a violin play? (Or a maqam player, or a singer, etc.?)

You may consider that this is a wrong way to ask the question, but I don't think so. I think that these questions are not enough considered, that they are rejected as apparently futile or unduly complex. But in the end, they cannot be avoided, and this is one of my main objections against the whole subject of "microtonal music" as it is too often presented. There is a lot to add, and I'd gladly continue the discussion if you feel like it; but I will not bother you further if you don't. — Hucbald.SaintAmand (talk) 13:00, 13 July 2016 (UTC)


 * Oh, it's an interesting discussion and point of view. I totally agree that even 12 et is an ideal. But I see it as like the mathematical idea of a cube or octahedron. You will never find a perfect cube or octahedron in our world. Always the edges will be unequal, the lines uneven, the faces not flat. But the concept of a cube is still very useful as a way of understanding and working with shapes. With twelve equal I think the nearest to it that you get in practice for real world instruments is with strings under low tension. It has been used for lutes since medieval times, I understand, for the same reason as it is used for guitars, because it's the simplest way to fret a fretted instrument. Even when theoretically it was pythagorean, I think lutes were tuned to twelve equal. I'd need to ask Margo Schulter to be sure, she's expert on this topic of medieval tunings. Certainly going back to Bach, then in his day guitars and lutes would be tuned in twelve equal.


 * And with singers - well you could sing in, for example, seven equal, or seventeen or nineteen equal. It might be an approximation and maybe sometimes the intervals would be closer to pure and maybe you intentionally sing in an adaptively tuned seventeen equal or whatever, but the structure of the chords and keys would show what you are doing, if you have seventeen distinct keys instead of twelve, for instance, you can't be in twelve equal.


 * As for the reason for using the word microtonal here - I think of it as an extension of the idea of microtones to the notion of an interest in microtonal distinctions of pitch - for instance a meantone fifth differs from a pure fifth by a micrtone and even though in meantone you don't have any very tiny steps, you do get those steps when you compare meantone with say pythagorean. And more generally I think that even twelve equal can be approached in a microtonal way when you think of it, for instance as the result of tempering both the fifths and the thirds of the lattice of all the intervals you get by combining pure fifths with pure major thirds. Yes most tuning systems have a finite number of tones, but some have infinitely many, e.g. most regular temperaments - if there are more than two intervals that differ from the octave and none of them are exact divisions of a multiple of octaves, then I think that's probably sufficient conditions to have infinitely many pitches in the tuning (would need to check). At any rate the two dimentional lattice of pure fifths and pure major thirds is one example of many with infinitely many pitches. Though normally of course they normally cut it off to a small subset of the lattice to make a scale for playing in. I've just today had an idea for a solution for this article so will start that as a new section. Robert Walker (talk) 14:11, 13 July 2016 (UTC)

Random break for convenience of editing
Yes, Robert, 12-ET is an interesting theoretical case. It has been around, conceptually, since Antiquity, since when one decided that the tone could be divided into two semi tones, implying that the two halves could be equal. But I don't follow you when you write that
 * It has been used for lutes since medieval times, I understand, for the same reason as it is used for guitars, because it's the simplest way to fret a fretted instrument.

Medieval lutes at times were tuned to a good approximation of ET, but their tuning was not a temperament. One made use of the 18:17 ratio, which amounts to 98.95 cents, but is an integer ratio nevertheless. There was no way to conceive a true tempered semitone, nor to geometrically represent it, before the late 16th century and Simon Stevin's continuous fractions (and later, of course, logarithms). Also, fretted instruments at times had double frets for the two enharmonic notes (i.e. they made a distinction between the diatonic and chromatic semitones). This was the case especially with larger instruments (viola da gamba); for shorter ones, strings were slightly pulled sideways while playing to obtain different intervals. ET was approximated on the lute because the frets, crossing all strings in one straight line, could not make the distinction between the two forms of the semitone (diatonic and chromatic), but certainly not because it was "the simplest way to fret": it certainly was not. And players often used expedients in order not to have to play in ET. This remained true even in Bach's time, when theorists (and Bach himself) advocated against ET because it would have reduced the difference between the various keys. To say that lutes were tuned in ET isn't really wrong, but it certainly is anachronic. [Margo Schulter has an expertise on medieval tunings, indeed, but not every scholar on this topic, me included, always agrees with her.]
 * ... you could sing in, for example, seven equal, or seventeen or nineteen equal.

Well, if we agree that 12-ET is more conceptual than real, the same can be said of any other ET. To say that a singer sings in 7-ET, or 17-ET, or 19-ET, really is a matter of conception. 19-ET, in particular, can so to say not be differentiated from 1/3-comma meantone (taking two units for the diatonic semitone): one hardly could say whether anyone sings in 1/3-comma meantone or in 19-ET, unless one knows her or his conception. There is a tendency, among, say, Xenharmonicians, to reduce everything to xx-ET and to precisely quantify microintervals. This is not wrong in itself, but it gives a rather distorted image or the realities of former times.
 * ... most tuning systems have a finite number of tones, but some have infinitely many, e.g. most regular temperaments ...

This again may be true conceptually, but it certainly was not thought so in former times because an infinite number of notes was unthinkable. (Infinite mathematical series again belong with continuous fractions and logarithms.) Meantone tunings were conceived at most with 17 notes, that is the diatonic scale + 5 sharps and 5 flats. The idea of double sharps and double flats is a recent one, and Xenharmonicians are way beyond that. Take a look at List_of_pitch_intervals: one could argue that no singable or playable interval is absent of the list; but to claim that sung or played intervals in real music do reflect those of this list is, to me, an aberration – and, for the ordinary Wikipedia readers, a misleading. Using integer frequency ratios implicitly claims that these are ratios between harmonic overtones; but what sense is there to imagine harmonic numbers in three figures and more, when hearing is concerned?

This all to say that I have problems, as a historian of music theory, to accept the Wikipedia articles about which we are speaking as vulgarizing concepts of earlier times. I don't mean that these articles are wrong, nor do I have excessive difficulties understanding what they do. I merely mean that I don't think they provide the simple explanations that the average Wikipedia reader is hoping to find. — Hucbald.SaintAmand (talk) 21:17, 13 July 2016 (UTC)


 * Thanks Hucbald. Okay this is going to be a long answer, hope you are okay with it. I'll start by correcting a number of things in what you say based on background of maths and interest in history of maths and in history of microtonal music.


 * Medieval lutes at times were tuned to a good approximation of ET, but their tuning was not a temperament.


 * Okay, interesting about the early lutes. First by "simpler" I meant simpler in terms of the fretting pattern there rather than simpler to tune. Because, as you say, the frets can go right across the fingerboard instead of being stepped or doubled. And of course they wouldn't have sounded as good to their ears aiming for the pythagorean tuning. I suppose it's a good point there, that they weren't really using it "as 12 equal" in the sense of fretting like that because they wanted to play in twelve equal, as with a modern guitar, but rather using it as a basis to try to play the lute in other tunings. So the intent is different from a modern guitarist, but still the default tuning of the lute if you just play without bending the notes is close to twelve equal.


 * There was no way to conceive a true tempered semitone, nor to geometrically represent it, before the late 16th century and Simon Stevin's continuous fractions (and later, of course, logarithms).


 * Well I can speak to the maths here as I'm a mathematician by training and with an interest in the history of maths. Interesting that they used 18/17. However, the maths of twelve equal conceptually only requires square roots and cube roots, and doesn't require logarithms or sequences. I'm talking here about concepts rather than whatever happened historically. The ratio of the frequencies of two notes an exact semitone apart is sqrt(sqrt(cube root(2))) = 1.05946309436 so 1.05946309436^12 = 2. Square roots are easy to do geometrically, by constructing a square and drawing its diagonal.As for the cube, you can do that using the exact solutions to the problem of Duplicating the cube, which was solved back in the time of Plato. So you could also solve that one in a conceptually simple way by first duplicating the cube, which gives you the cube root of 2 by a complicated geometrical construction using for instance marked rulers that you slide into position against other lines you have drawn. Then you use another geometrical construction for the square root, and do that twice. You now have two lines with their lengths at a ratio of 1 : twelfth root of 2, and using that you could go on to construct your fretting pattern.


 * So they had the mathematical capability to find the exact spacing of a fret for twelve equal. Whether anyone did do it this way of course is a matter of history rather than theory. So this is just a correction for the maths, not the musical history.


 * While discussing maths, same also applies to the idea of infinitely many pitches:


 * This again may be true conceptually, but it certainly was not thought so in former times because an infinite number of notes was unthinkable


 * The idea of prime numbers and divisibility goes back to Euclid's elements, and ratios back then also Fraction_(mathematics), and they had the concepts to understand infinitely many notes, again whether they applied those concepts to musical pitches is a matter of history rather than maths, but the had the mathematical tools for sure. That's different from modern ideas of infinite series which they didn't have. But that's not needed to understand this idea of infinitely many fractions.


 *  Meantone tunings were conceived at most with 17 notes, that is the diatonic scale + 5 sharps and 5 flats


 * The observation that 31 equal approximates meantone almost exactly goes back to Huygens in 1661, published in 1691, after the invention of logartithms. 31 equal unlike 12 equal would have been just about impossible to construct exactly before logarithms. But Vincento's Archicembalo approximated 31 equal and Huygens says the idea is not his, but refers back to Saliinas' report on Vincento's Archicembalo as the origins of this idea that quarter tone meantone is approximated closely by 31 equal. The Archicembalo was an instrument that could be tuned to play 31 almost equal pitches to the octave. So the idea of playing in ETs other than 12 et goes back to 1661 at least but arguably back to the Archicembalo. Huygens himself built a "mobile keyboard" consisting of 12 keys per octave that could be moved around to play different subsets of a set of 31 strings or pipes tuned to 31-et, so permitting play in all 31 keys of 31 equal. Details of how it worked not very clear but is clear it was an actual physical instrument.

And there have been many other experiments over the years. Notable recent one of course, Adriaan Fokker' organ - he was composing for 31 equal back in the 1940s?


 * Helmholtz's "On the Sensations of Tone" published in 1885 has a long section describing many instruments of his day to explore many tones to the octave with modulation. He was particularly interested in 53 equal which very closely approximates just intonation, and this is his description of Paul Whites harmonium designed to play in 53 equal


 * The Orthotonophonium MIM Orthotonophonium.jpg dating back to 1914 by Arthur von Oettingen could play in 53 equal or 72 equal.


 * So there's a long history of keyboards in many equal temperaments. In Turkish maqam music I understand from talking to Turkish musicians / composers that they often understand their music within a theoretical framework of 53 equal also, don't know how far back that goes historically - though nowadays they also explore other ETs for them.


 * So that's by way of background, now to your main points.


 * First, of course I don't expect all experts on medieval tunings to agree, as in any academic subject there's going to be a wide debate. It might be interesting if Margo Schulter were to join this debate. Anyway to make a few more comments on your main points.


 * Take a look at List_of_pitch_intervals: one could argue that no singable or playable interval is absent of the list; but to claim that sung or played intervals in real music do reflect those of this list is, to me, an aberration – and, for the ordinary Wikipedia readers, a misleading. Using integer frequency ratios implicitly claims that these are ratios between harmonic overtones; but what sense is there to imagine harmonic numbers in three figures and more, when hearing is concerned?


 * Yes but it's the same as the perfect cube idea. Those more complex intervals there show interrelationships of pitches. For instance the 32805 : 32768 or Schisma - that's 38·5 : 215 well from that you can see that you get the schisma by going up eight pure fifths then by a major third and reducing to the octave. So - there's no way that anyone is going to sing a  32805 : 32768 (1.95372 cents) which could be distinguished by ear from a  32804 : 32767 (1.95378 cents) - even in long held chords listening to beating partials, it would be hard to make such a fine distinction. But they can tune by eight pure fifths, and then by a major third reducing to the octave all the way. And if you then reason about what they are doing mathematically, it's very useful to have this notion of a Shisma. It's like that in nearly all except the simplest of applications of maths, you use ideas that just can't be measured that exactly. For instance the area of a disk of diameter 1 is 3.14159265359 and even that is not at all exact, you can go on for millions of places and it's still not exact. So you use approximations in real life, but when you do the maths you don't think of pi as 3.14 or even 3.14159 but as that infinite decimal. It's like that. So, much as one might want to, you can't tell the theorists working on microtonal theory that they have to use approximate numbers that can be distinguished by ear. They wouldn't be able to do their maths if you did that. To reason about tunings they have to be able to use these mathematically defined exact ratios. Nobody claims they can be distinguished by ear.


 * There is a tendency, among, say, Xenharmonicians, to reduce everything to xx-ET and to precisely quantify microintervals. This is not wrong in itself, but it gives a rather distorted image or the realities of former times.


 * Rightio. I think it's important to be historically accurate. As an example in the article on the Archicembalo just to pick up on something I saw just now, where it says "The most important was the extended quarter-comma meantone temperament, which given such a wide gamut of fifths becomes almost exactly a system of 31 equal divisions of the octave " - it would help to add something like "This was recognized by Huygens in 1661, published in 1691, He says that after making this discovery, he found an earlier mention of the idea in Salinas' report on Vincento's Archicembalo". Indeed I'll add that now, it should help. I.e. to not just say that it is almost exactly 31 equal, but also to say who was first to recognize that it was almost exactly 31 equal. Did Vincento himself recognize this? Well seems we don't have evidence that he did, of course correct it if we do - but do have evidence that Salinas recognized it as such. Might well be that he did also, it's quite a natural thing to notice that the keys are more or less equally spaced, but to be accurate, say what we actually know for sure.


 * I'm all for making the articles more historically accurate!


 * This all to say that I have problems, as a historian of music theory, to accept the Wikipedia articles about which we are speaking as vulgarizing concepts of earlier times. I don't mean that these articles are wrong, nor do I have excessive difficulties understanding what they do. I merely mean that I don't think they provide the simple explanations that the average Wikipedia reader is hoping to find. 


 * Right I agree on simple explanations. I've seen quite a few articles here that have complex algebra and other advanced maths, which is all very fine, that needs to be included for the theorists. But they sometimes lack simpler explanations. Indeed this page itself is a case in point. It never explains that 5-et is the result of tempering out the diatonic semitone and that 7-et is the result of tempering out the chromatic semitone. That's simple to say, and would make the whole article much clearer. If I rewwrite it as an article on linear tempermanets, then I'll do that. That's actually something I might be able to help with. I'm a science blogger, in my spare time, and I write a lot on various topics, mainly space, but also maths, and music at times. And people say that I'm good at this sort of thing. So - I'm not expert on the theory or the history, but I can understand at least some of the maths (except when it gets very algebraic, then I get lost) and am quite good at explaining what it means in ordinary language, once I understand it. I think that should be another project to take on with the new WikiProject. I'll add that to the project description.

Robert Walker (talk) 10:10, 14 July 2016 (UTC)


 * Just to add, may interest you, this author says that the 18/17 rule is actually more accurate for a lute than the twelfth root of 2.
 * "...18:17, which happens to make a very good prescription for placing the frets down the neck of a lute for equal temperament. It puts the octave shy of the string's midpoint by some 1/3 of 1% of the total length (comparable on a tenor lute to the width of the fret itself). This might be considered a defect from a certain theoretical point of view, but in reality 18:17 works better than twelfth root of 2 as the latter makes no allowance for the string's greater tension when it is pressed down to the fret. On a good instrument (that is, with a low action) the 18:17 rule renders the string just about long enough to compensate"


 * Happened to find it when looking up something to do with the archicembalo which is the main subject of the article. Robert Walker (talk) 23:06, 14 July 2016 (UTC)

Random break 2
To Robert Walker. Robert, I'll organize my answer around many quotations of yours. First:
 * And of course they wouldn't have sounded as good to their ears aiming for the pythagorean tuning.

Whether medieval musicians aimed for the Pythagorean tuning is not clear. It is true that the theorists mentioned only this tuning, but one may wonder whether singers really sang Pythagorean thirds, when music began to use thirds. As early as the 14th century, keyboardists had found what has been called (by Helmholtz or Ellis?) the "schismatic change" (Schismatische Verwechslung; it really is a commatic change) to avoid Pythagorean thirds, tuning the black keys as Pythagorean flats and using them as sharps in major thirds of 388 cents. This is documented, among others, in several papers by Mark Lindley. The problem with lute tunings may not have arisen long before, so that it is not at all sure that lutenists ever aimed for Pythagorean tuning.
 * So they had the mathematical capability to find the exact spacing of a fret for twelve equal.

Indeed, they could have calculated or constructed the square root of the square root of the cubic root. But I think it is quite known that they didn't. It certainly is known that Simon Stevin was the first to approximate the 12th root of 2 by continuous fractions, and Napier's logarithms were soon recognized, in the 17th century, to at last allow calculating it. This all produced a major change in "music" itself: it had been up to then another name for the science of ratios of integers and, as such, had belonged to the quadrivium; it was a science of discontinuous quantities. But the new mathematics changed that, music lost its place in the quadrivium (which disappeared altogether), it had to seek its models in the sciences of language (mainly rhetorics), and eventually was rejected from university curricula. There were other reasons, of course, but the developments of mathematics were part of it.
 * Vincento's Archicembalo approximated 31 equal and Huygens says the idea is not his, but refers back to Saliinas' report on Vincento's Archicembalo

I do believe that the situation in 1661 was drastically different from that in 1555. Vicentino apparently was not aware that 31-note meantone approximated 31-ET. So far as I am aware, Salinas knew of an instrument giving 31-tone meantone, but did not know its maker; there is no "report on Vicentino's Archicembalo" by Salinas, nor any mention of it in Huygens who, in addition, apparently did not know Vicentino's book. (Be aware of this before you modify the Archicembalo article!) Huygens does refer to Salinas, who does refer to an Italian instrument, but neither mentions Vicentino. Vicentino himself did not think that his instrument could serve for extended meantone (i.e. an extended range of tonalities in meantone), but conceived it for playing microtones (1/2 or 1/3 semitones) for which he devised a special notation. Other similar instruments of the time more often aim at extended just intonation, which is quite a different matter. (Extended meantone would have made it possible to play "tonalities" that were very close in pitch and identical in all other respects. Vicentino aimed at playing what he would have considered "enharmonic tonalities", i.e. modes or keys returning to what he believed was the enharmonic system of the Greek.)
 * He was particularly interested in 53 equal which very closely approximates just intonation, and this is his description of Paul Whites harmonium designed to play in 53 equal

This is not by Helmholtz, it is found in the additions by Ellis in his English translation. But Helmholtz does mention the 53 division (e.g. p. 328 of the translation). This is Holder's (18th c.) or even Mercator's (late 16th c.) approximation of the Pythagorean tuning, with 9 "commas" (53th root of 2) in the tone, 5 in the chromatic semitone, 4 in the diatonic one; it had been known, conceptually, in Greek Antiquity. But this was theoretical speculation, which left few traces in actual music.
 *  In Turkish maqam music I understand from talking to Turkish musicians / composers that they often understand their music within a theoretical framework of 53 equal also

I don't think this goes back any farther than the 18th century (I could find the source, but not just now). The Cairo congres of 1932 advocated 24 equal quarter tones for Arabic maqam music, and I have had Arabic students who were convinced that this went back to the early Middle Ages...
 * For instance the 32805 : 32768 or Schisma

The schisma was described long ago as the difference between the Pythagorean and the syntonic commas (this is not a definition, because schisma really means any very little interval), which can be calculated in a variety of ways. I am not convinced that 32805:32768 (or 3^8*5 : 2^15) makes this so much clearer.
 * It never explains that 5-et is the result of tempering out the diatonic semitone and that 7-et is the result of tempering out the chromatic semitone.

I am not sure of what you mean here – as a matter of fact, I don't understand what "tempering-out" means. I suppose that what you mean is that 5-et is a temperament where the 5ths are tempered so that they reach unison in the end, instead of a diatonic semitone; that is, 5-et is a "positive" temperament where the 5ths are augmented by 1/5 diatonic semitone. I feel far-fetched to call this "tempering-out".
 * Just to add, may interest you, this author ...

This author, Mark Lindley, is the one I mentioned above re the "schismatic change": you'll find his papers on this in his (long) list on Academia.edu. He is one of the best scholars on the history of tunings and may usually be trusted.

My point, il all this, is that the mathematical discourse about microtones and tunings appears to afford all kinds of exact quantifications, applied to a historical reality that never had any use of this kind of exactitude. One may get the illusion that mathematical definitions are more rigorous and therefore better than the earlier fuzzy definitions, but music, like any other language, cannot be dealt with in such terms. Tuning appears to be of paramount importance because it would allow an exact definition of the various notes, of their exact frequency, etc. But notes are first of all semiotic categories – while even these are today mistaken for classes (e.g. in pitch-class set theory). A semiotic category fully escapes mathematical definition. — Hucbald.SaintAmand (talk) 11:39, 15 July 2016 (UTC)


 * To : Hucbald, interesting information! First, I have already "been bold" and edited the Archicembalo article as I thought it's best to do something right away to correct the impression that Vicentino thought in terms of 31 equal. From the cites I brought up, it seems that his 31 tone system must have been somewhat uneven as he says that you get a purer major third by combining notes from the front keyboard with the back keyboard which of course would not be true in an exact 31 tone system. That seemed like pretty conclusive evidence to me, so I agree it can't have been tuned originally to either extended meantone or 31 equal. It's cited there as the opinion of scholars of course, Archicembalo. Do correct any errors you see in what I added of course, or add extra cites / details. I also commented about the addition on the article's talk page.


 * On tempering out - I mean that in the same way that quarter comma meantone "tempers out" the syntonic comma, and twelve equal "tempers out" the pythagorean comma. In the same way 5 equal "tempers out" the pythagorean diatonic semitone which you can think of as a kind of a large comma, and 7 equal "tempers out" the pythagorean chromatic semitone. Perhaps it is better to say "tempered by the diatonic semitone"?? It's the same process anyway, whatever you choose to call it. A tempering that removes all distinctions of the diatonic semitone, or the chromatic semitone respectively.


 * With the schizma, one can choose whether to use it's numerical value or not. It's not so easy to see that it's the difference between the syntonic comma and the pythagorean comma. But from the 3^8*5 : 2^15 =  (3/2)^8*5/4 : 2^5 you can do a quick calculation, even do it in your head, to see that it's the "difference between 8 justly tuned perfect fifths plus a justly tuned major third and 5 octaves".  I think the actual numbers themselves are not so useful, but the prime factorization of those numbers is very useful, indeed it might be an idea if the pages that define these things should add the prime factorizations in brackets. For instance the page on the Schisma I think could benefit from having the prime factorizations of all the commas given - perhaps as an insert to the right to not disturb the flow of the text. After all how often do you want to know that a Schisma is 32805 : 32768? While you'll often want to know that it is  3^8*5 : 2^15. I can understand that it can make the page look overly mathematical given that there are many readers who are probably not that comfortable with exponents, it could be done as a float to the right  Schisma = 32805 : 32768 =  38*5 : 215 1.95372 cents  Syntonic comma = 81:80 = 34 : 5·24 21.50629 cents  Pythagorean comma = 531441 : 524288 = 312 : 219 23.46001  cents   Pythagorean diatonic semitone 256 : 243 28 : 35 90.225 cents (  Pythagorean chromatic semitone 2187 : 2048 37 : 211  90.225 cents ) which would list all the commas mentioned in the article for easy reference. Just an idea. and do that with all the commas mentioned in the article for easy reference. I did it here with html divs, so quite techy, could make a template to do this, which would make it easy to do. Could also add the cents values for each one.


 * On your last point I totally agree, that the modern exact definitions become anachronistic if applied backwards to earlier times to musicians and theorists who didn't think in those terms. It depends on the context. For instance in maths, then it does make sense to use modern algebra to describe the algebra of early mathematicians even though they used much more clumsy systems in words. Except for instance when you want to talk about why they treated the quadratic as several separate cases, you might then talk about how they expressed it in their own language.


 * So I think here also, it helps the modern reader to use the modern terminology, so they understand it more quickly and easily. Even for Vicentino's system, for a modern reader who is familiar with equal temperaments, you get a good first idea of roughly how the notes were pitched by saying that they are approximately in 31 equal and then to talk about deviations of the notes from extended meantone and 31 equal even if Vicento never thought of them in those terms. But then you also need to be historically accurate and explain to the reader that Vicentino didn't think those ways. I've added a few extra things to the Examples of things we could do to the project proposal:


 * "Make the articles historically accurate For instance in the article on the Archicembalo they say that his tuning approximates 31 et - which is a modern observation - but when was that recognized historically? Turns out that it was recognized at least by 1661, just added a cite for that, did Vincento recognize it himself? One project could be to go through the articles checking matters of what was understood historically at the time, for historical tunings.
 * "Add simple language explanations Many of the articles describe concepts in complex mathematics and algebraic language. That's fine and necessary for those who are interested in the advanced maths of tuning theory. But often the ideas are not explained in ordinary language. We could work on providing accessible ordinary language explanations to the articles, for instance in the lede and at the start of a section, explain what it is about in ordinary language before the maths."


 * On the Maqams, yes, interesting. I know they had some quite elaborate system going way back, a Turkish musician explained it to me at some point but don't think it was 53 et. I don't think they use the 53 equal or extended pythagorean as a reference for intonation, not like Western music where the singers and players may actually try to play their closest approximation to twelve equal. For them, I think it is much more of a theoretical framework. The tuning of the "same" note varies anyway depending on context.


 * It's a little bit like that thing about the 31 equal being used for Vicentino's system, you can say that his major thirds vary by (say) -0.2 comma to + 1/3 comma from extended meantone / 31 equal and for a modern reader that helps you to understand more about how it worked although it's not at all how he thought of it. You can use many different frameworks as a reference, and with all the intonation changes you'd obviously need more than 53 notes to make fixed pitch instruments to play Turkish music with those subtleties and some do.


 * Anyway I totally agree that the articles here should reflect things like this. And - I think is legitimate to say that singers and instrumentalists do often try to play and sing in twelve equal as best they can, and you can do the same in other systems, may recognize that they play different tunings in sustained chords or expressively, or may not even notice they do it. And that we have that mathematical background of 12 equal can also be used to study how much they vary away from 12 equal too, not in the sense that 12 et is the correct tuning, but just as a way to describe what is happening reasonably precisely, so same also with other tunings. And sometimes a system is used in both ways as happens in twelve equal, and sometimes it is used only as a theoretical framework but no attempt made to sing it exactly, as in the 53 et maqams. Robert Walker (talk) 13:02, 15 July 2016 (UTC)

Idea - let's just move this to "Linear Temperament"
I've just had an idea. Wikipedia doesn't yet have an article on linear temperaments, it just redirects to regular temperament. So - as an interim measure, why not make this into a stub article on linear temperaments? I think I can do that quite easily, just rewrite it, explain what a linear temperament is. Then can say that for tempered fifths, you get twelve tone tunings but if you temper the fifth flat enough you will temper out the diatonic semitone and the result is five equal, and if you temper it sharp enough, you temper out the chromatic semitone, resulting in seven equal. Then say that Andrew Milne has coined the word "Syntonic temperament" to covert the continuum of tunings in this range. Whether that is notable enough to be mentioned can be a matter for further discussion but as an interim measure can do it like that. And I can also say that tunings with fifths sharper than pure are called schizmatic temperaments and flatter than pure called meantone temperaments - and add a "citations needed" for the schizmatic temperaments so hopefully someone can come up with the citations you've been looking for there.

Then can add a "please expand". As is, it is only mentioning a tiny subset of the linear temperaments, and experts can come in later on with temperaments based on generators other than a fifth, I know there are lots of those but am not expert on them :).

~The advantage is as an interim measure it involves hardly any work and we can do it a bit at a time. Even if the other articles still say "syntonic temperament" well after clicking through at least they will see syntonic temperament mentioned further down the page. And they won't puzzlingly find themselves on the meantone page which I think would be very baffling, if you are there already, or if for instance you are on some other schizmatic temperament type tuning, click on syntonic temperament and you end up thinking it is a form of meantone which might be very wrong. Then after doing that can go through and replace occurrences of "syntonic temperament" by "linear temperament" and at every point in the editing process then wikipedia's articles are linked together in a reasonably sensible way. What do you think? Robert Walker (talk) 14:22, 13 July 2016 (UTC)

New idea - let's move it to "Regular Diatonic Tunings"
Margo Schulter has just suggested this off wiki. This is both
 * A logical name, combining "Regular" with "diatonic"
 * It's also used in the literature, see for instance chapter heading here:

It would cover the range from 5-et to 7-et, but not including the two end points. The word "Regular" here is understood in the sense of a homomorphism map from the pythagorean diatpnic such that all intervals of the same type are tuned the same, wherever they occur in the tuning. As in Dynamic Intonation for Synthesizer Performance by Benjamin Frederick Denckla (1995). In other words any scale consisting of "tones" and "semitones" arranged in the sequence TTSTTTS and adding up to the octave with all the Ts tuned the same way and all the Ss tuned the same way counts as a regular diatonic tuning.

This also includes to all linear temperaments within Easely Blackwood's "Range of Recognizabilty" in his "The Structure of Recognizable Diatonic Tunings" for diatonic tunings with the fifth tempered to between 4/7 and 3/5 of an octave, with major and minor seconds both positive and the major second larger than the minor second, though his "range of recognizability" is more restrictive than "regular diatonic tuning", for instance requiring the diatonic semitone to be at least 25 cents in size. See Carlo Serafini's review for a summary.

This has the advantage that we need minimal rewrites of wikipedia, just replace "syntonic temperament" by "regular diatonic tuning" everywhere and rename the article, and is clearly noteworthy enough to deserve an article of its own I think. - any thoughts on this? If there are no objections or other suggestions, I think I'll follow WP:BOLD and just do it, and rewrite the article also to make it all clearer. Robert Walker (talk) 12:04, 19 July 2016 (UTC)


 * I've "been bold" and made the move. Also marked it as "work in progress", will work on it some more over the next day or two. Robert Walker (talk) 13:38, 24 July 2016 (UTC)

"When the fifths are a little flatter than the 700 cents [...] we are in the region of the historical meantone tunings"
There is only one historical or logical definition of a "meantone", unless I am mistaken:
 * The mean tone is the mean of the major third – which is achieved whenever the four fifths producing the major third are equal.

I see no historical or logical reason to limit the definition of (historical) meantone tunings to "negative" temperaments, i.e. those tempering the 5ths to flatter that 700 cents. Equal temperament is a meantone tuning, as is Pythagorean tuning and also the "positive" temperaments, producing 5ths sharper than 702 cents.

I very much dislike the idea of "tempering out" (or "distributing") the syntonic comma, which to me is meaningless. No historical temperament ever tempered anything else that the fifth. And a "meantone tuning" with 5ths larger than 700 cents widens the syntonic comma... A circulating 17-note "Pythagorean" tuning needs 5ths of 706 cents; it is "Pythagorean" in that the diatonic semitone is smaller than the chromatic one; but the major third still consists of two tones. Etc.

Hucbald.SaintAmand (talk) 20:30, 24 July 2016 (UTC)


 * Okay, first yes, in theory you could have meantones that go smaller than 1/11 comma, for instance 1/12 comma has a fifth of 700.2 cents


 * Yes, it's an awkward phrase when you think about it. Because the word "meantone" properly only applies to "quarter comma meantone" because that's the only one that achieves a musical meantone between the 10/9 and the 9/8 as sqrt(5/4). If you apply the word strictly,none of the others are meantones at all, not even 1/6 comma.


 * But third comma, sixth comma etc meantone is certainly established usage. As an example, here is a google scholar search for "sixth comma meantone". We have to go by established usage in wikipedia, and this is surely established usage. And historically, Zarlino first describes 1/4 comma and 1/3 comma meantone in 1571, and 2/7 comma meantone in 1558. I should add that to the article to make this clear, that it's not a modern system but goes back a lest to the sixteenth century. See An historicla survey of meantone temperaments by Mark Lindley


 * You are right that in the range from 700 cents to 3/2, then you could describe the tunings as meantones, mathematically at least, if it means that you temper the fifth by a positive fraction of a syntonic comma. But are there any historical mentions of tunings that temper the fifth by less than 1/11 of a syntonic comma?


 * For fifths sharper than 3/2, then the syntonic comma would need to be tempered in a negative direction, so you'd have the likes of -1/6 comma meantone. It's certainly possible to make such a tuning - I have a meantone scale generating applet on my website that accepts negative values. So, is easy to calculate, -1/6 comma meantone is 0, 138.8, 211.1, 283.4, 422.2, 494.5, 633.2, 705.5, 844.3, 916.6, 988.9, 1127.7, (1200). But that's not established convention. Have you ever seen a linear temperament defined using negative fractions of a syntonic comma? Instead for linear temperaments with a fifth sharper than 3/2, the tuning is described in terms of fractions of a schisma.


 * I can see why you might find this awkward terminology. But on wikipedia we have to go by established usage rather than by what we prefer personally or find more logical.


 * BTW thanks for your recent edits which improved the article. Yes that was a mistake to present the chain of fifths without making clear to the reader that they are reduced to the octave, making an ascending fifth equivalent to a descending fourth. I wasn't sure what to do about your sentence about the semitone as an alternative generator of twelve equal. It didn't seem relevant here as you can't use a semitone to generate a regular diatonic tuning, or indeed, any other twelve tone tuning. You can however use the fourth as an alternative generator so I mentioned that. Hope you are okay with those edits, also did some formatting improvements. Thanks, Robert Walker (talk) 21:49, 24 July 2016 (UTC)

Robert, you assume that when I define the mean tone as "the mean of the major third", I mean "the mean of the pure third", but that is not the case. I mean that the mean is the mean of the particular major third of that particular temperament. 12-ET is a meantone because the tone (200 cents) is the mean of the tempered third (400 cents). [This means quite a lot of means, but I hope you can understand what I mean ;–))]

J. Murray Barbour writes (Tuning and temperament, p. 31): "Strictly, there is only one meantone temperament. But theorists have been inclined to lump together under that head all sorts of systems". And he later comes to equate meantone temperaments with what Bosanquet called "regular" temperaments in 1876, namely the temperaments with only one size of fifth. This also is what I understand by "meantone temperament". Barbour further describes 2/7-comma and 1/3-comma, then discusses the possibility that the temperament was progressively reduced from 1/4-comma to 1/11-[syntonic] comma, which is equal (he says) to 1/12-Pythagorean comma.

[By the way, the name of Bosanquet should probably be given in the Regular Diatonic Tunings article, as the originator of the term "regular".]

I always considered that the range of historical meantone temperaments extended from 1/3 to 0, i.e. 5ths from 695 to 702 cents. The possibility of negative temperament (fifths wider than pure) is a bit farfetched, yet a closed 17-ET, which strictly speaking may be considered a meantone and in any case a "regular" temperament, has 5ths of 706 cents. It is described by Sauveur as the "Arabian scale" and indeed was almost exactly al-Farabi's system in the 14th century. The 22-note division, which has been equated with the "Hindoo scale", is a regular temperament with 5ths of 709 cents. Note that not all multiple divisions produce "negative" 5ths, as Vicentino's 31 division evidences: it is a 1/4-comma meantone. But if 31-ET can be described as a meantone, I don't see why 17-ET or 22-ET (or any ET, for that matter) could not.

As to my mention of the semitone as a possible generator of the 12-note scale, it results from a reflection on the idea that the generators are the numbers that are prime to 12: 1, 5, 7 and 11 – the semitone and the fourth, and their inversions. Indeed, 12 has the property to have that many dividers, which makes it an excellent choice for combinatorial possibilities. But I admit that it is not extremely helpful here (it is helpful for a reflection on modes of limited transposition in 12-ET, because these obviously are based on the divisions of 12 in equal parts).

Hucbald.SaintAmand (talk) 06:40, 25 July 2016 (UTC)

For an image of my conception of meantone temperaments, see, which also appears in Meantone temperament. Hucbald.SaintAmand (talk) 08:09, 25 July 2016 (UTC)


 * Hucbald, Okay, I see what you are saying, that any regular diatonic scale is a meantone in that more general sense that the tempered major third is equally divided into two equal tones. However, though that is logical, it doesn't correspond to established usage. As an example, 17 equal, with a fifth of 705.884 cents, so tempered by (705.884-701.955) or 3.929 cents wider than just, is -1/5.47 meantone but I don't think anyone calls it that, and it would also be almost exactly -2 schisma but again that doesn't sound right either.


 * So first, I'm reluctant to edit that section to say that historical meantones extend to 0 comma meantone without a historical example of a less than 1/11 comma meantone. If you can find one, then for sure we can edit it to say so, as that will be historical proof that they used it in that sense.


 * I understand the awkwardnes. As a programmer and mathematician, I often encounter established usage in music theory and practice that doesn't sound logical. In the area of rhythms for instance, then it seems established usage amongst music theorists (this is all a bit of a generalization but usually), that a "polymeter" is any kind of rhythm involving several rhythms playing at once with different measures lengths, or the beats different, or both, while a polyrhythm is a measure preserving polymeter (e.g. 4:3) and polytempo is used more generally than a polymeter as the most general term, including tempo varying independently for each rhythm, while they don't seem to have a separate term for a beat preserving polymeter. But in popular music (usually), then "polyrhythm" refers to a beat preserving polymeter in the theorist's sense, e.g. 3/4 : 5/4 with both rhythms having the same size of quarter note, and they don't use the word polymeter, and don't have a separate name for what the theorists call a polyrhythm. Meanwhile classical musicians (often at least) refer to a beat preserving polymeter as a "polymeter" and a measure preserving polymeter as a "polyrhythm", almost the opposite way around from popular musicians. It's just a mess, and in the help for my polyrhythm/ polymeter program I simply have no possible "good choice" for terminology that would please all the users of the program. There isn't any logical reason for any of this. It is just what has become established practice.


 * But what can we do? We can't make up new terminology for wikipedia. However, I realize that I had a misunderstanding about schismatic temperaments. They are really for a narrow zone with the fifth temepred in the same direction as the syntonic comma but by much less, for instance tempered by 1/8 schizma to achieve a pure 5/4 in a descending chain of eight fifths. Sorry about that. I'll edit the article accordingly. So if I understand correctly (will check this), the tunings are referred to as meantones up to 12 equal, then as schizmatic in a narrow range below 3/2, although as you say technically those tunings are meantones as well.


 * I'm not sure now what the terminology is for fifths sharper than pure, as neither meantone nor shismatic seems right for them. They are quite common. Other popular temperaments with the fifth sharper than pure apart from 17 equal include 22, and 15 tone equal temperament. While for the fifth tempered less than pure we have all the multiples of 12, 19, 31, 53. I'll ask for help from the experts on the xenharmonic alliance about all this, and also ask Margo. Once the microtonal project is underway (hopefully), this is something we could raise on its talk page but we don't really have anywhere on wiki to raise larger questions like this at present, which is my motivation for the new project. Anyway meanwhile I'll fix the article first, then ask for help about this more general terminological question, and come back to you on it. Robert Walker (talk) 09:36, 25 July 2016 (UTC)


 * Robert, I am not entirely sure about schismatic [skhismatic] temperament, but I think that these terms originated in Helmholtz (Tonempfindungen) and in its translation by Ellis (Sensations of Tone). The affair is rather confused, but it seems that for some reason the tunings concerned make the "shismatic shift" by which the (major or minor) third aimed at by the series of tempered 5ths is taken by enharmony, C-F♭ for C-E or C-D♯ for C-E♭. You'll find many comments about this in the Talk:Schismatic temperament page where, as usual, I played my role of the bull in the China shop. The term apparently refers to the fact that these intervals, diminished 4th and augmented 2d respectively, if tuned in Pythagorean tuning, are a schisma distant from their "just" enharmonies, i.e. 5/4 and 6/5. Similarly as in a meantone temperament the major third is aimed at by four 5ths, in a schismatic temperament the diminished fourth is aimed at by eighth 4ths; etc. This, I trust, provides a much clearer definition of what a schismatic temperament is. Once again, see Talk:Schismatic temperament for more details. — Hucbald.SaintAmand (talk) 18:48, 25 July 2016 (UTC)
 * Note also that the Talk:Schismatic temperament page explains why I hate the idea that schismatic temperament "tempers out" the schisma, or meantone temperament "tempers out" the comma. Indeed, 1/4-comma meantone results in that the particular comma between the Pythagorean and just major third vanishes, but part of the comma between the Pythagorean and just minor third is still there: one certainly cannot say that "the syntonic comma" has been "tempered out", one should at least say which syntonic comma. And this soon results in a mess. The only interval that a temperament tempers is the 5th! Hucbald.SaintAmand (talk) 19:00, 25 July 2016 (UTC)


 * Hucbald, yes again I can see what you mean. When they say that the 81/80 syntonic comma is "tempered out" in quarter comma meantone, what they really mean is that the 3 in the 3^4/(5*2^4) gets replaced by (5*2^4)^(1/4) = 2.99069756244 (decimal value rather than cents value) - the 5 is left unchanged as is the 2, so in quarter comma meantone the tempered syntonic comma is (2.99069756244...)^4/(5*2^4) = 1/1, or 0 cents So that's what they mean, it's really a syntonic comma with a tempered 3 replacing the 3, rather than a tempered syntonic comma, but it's become established usage to call that the tempered syntonic comma. In other more general situations, both the fifth and the third may be tempered so a tempered syntonic comma doesn't have to have only the fifth tempered. That's what they mean when they talk about how various commas vanish in various equal temperaments in the xenharmonic wiki. They mean that if you replace the 3 by the tempered version of that number in the tuning, the decimal value for the fifth, and replace 5 by the decimal value for the major third (or closest to it in the tuning) etc that the result gives you 1/1. Anyway so that's established usage again, but a bit awkward I agree. It does give you useful information. If a equal temperament has the syntonic comma "tempered out" that means that if you go up by four fifths (reduced to octave) you get to whatever is the major third in that system. Similarly if the schisma is tempered out then going up eight fifths (reduced to the octave) takes you to whatever is the et's best representation for a just minor sixth etc, so from knowing which commas are tempered out, you know a lot about how the system works. So it's a useful concept, the theorists definitely need it and can't do without it or something like it, but the terminology is somewhat awkward I agree. But maybe this explanation can help make a bit more sense of it? Robert Walker (talk) 19:38, 25 July 2016 (UTC)

Fifths discussion - random break for editing
( - just to say finished editing this post. Robert Walker (talk) 14:38, 26 July 2016 (UTC))

Hi Hucbald, been talking about this to Margo so far, anyway she agrees with you about the meantone being a logical name for the entire gamut of regular diatonic tunings. She prefers to use the word "eventone", I think to avoid confusion with the historical usage. Anyway she suggests, though I'd say we can probably use this informally on the talk pages only, not for the wikipedia articles unless we can establish that it is used in WP:RS for wikipedia - that one could generalize the idea by using commas that require fifths to be tempered wide instead of narrow. E.g. etc.
 * "1/4-(896/891) meantone, or "1/4 undecimal comma meantone" (if undecimal comma is understood in this way) for wide fifths at 704.377 cents for a pure 14/11 major third.
 * "1/3-(352/351) meantone" or "1/3-tredecimal comma meantone" (if tredecimal comma is understood n this way) for wide fifths at 03.597 cents for a pure 13/11 minor third
 * 1/14 (14680064/14348907) meantone for a pure 7/6 minor third from +14 fifths, or two apotomata

In the xenharmonic wiki they also have a distinction between positive and negative temperaments, which puzzled me at first, as positive there means the fifth is wider than 700 cents, not 3/2. Margo says that she uses those words also though and gave an explanation that made sense to me. She tends to say tunings with fifths larger than 700 cents are "positive," in the sense that twelve fifths exceed 7 octaves (version of the pythagorean comma using the tempered fifths is positive), and corresponding sharps are higher than flats (e.g. C-Db-C#-D). That is true throughout the regular diatonic range of 700.0-720.0 cents

On the meantone / shizmatic then she says that the 700 cents is a logical transition point between the two as it is the point where a regular or meantone mapping of 5/4 (+4 fifths) and a schismatic mapping (-8 fifths) yield an identical 400 cents.

Then on the schismatic temperaments, for her, they occupy a narrow region around the 3/2.


 * from 700 cents up to 3/2 for the normal 5 limit 32805/32768 schisma ( 3^8*5/2^15) or 1.954 cents.


 * Then beyond that you can optimize ratios of 2-3-7 using fractions of the "septimal schisma" of 3.804 cents 2^25/(3^14*7) or 33554432/33480783.

It means that for example a fifth at 702.227 cents (0.272 cents or 1/14 of this septimal schisma wide) produces pure 7/4 minor sevenths from -14 fifths and large 8/7 tones.

BTW, hope you don't mind a minor correction. I'm saying this because I know if I was in your position I would want it and I think most academics are the same.

On the 17 tone tuning, she thinks you are probably thinking not of the ninth century theorist Al-Farabi (c. 870-950), but of Safi al-Din al-Urmawi (one possible transliteration), c. 1216-1294.

If so, it was a 17-note Pythagorean tuning, or alternatively, another system by the same theorist setting the Wusta Zalzal at 72/59 or 344.7 cents which is similar to Ibn Sina (c. 980-1037), based on a Wusta Zalzal at 39/32. None of these systems is close to 17-ed2, and they involve chains of just 3/2 fifths -- a single chain with Safi al-Din's Pythagorean scheme, or more than one chain in his alternative scheme with the Wusta Zalzal or middle third at 72/59.

I'm sure she'd be delighted to give more detail if you want to email her about it :).

BTW I know it's a little awkward that I'm kind of relaying what Margo says to you, but I think she is wise to not get too involved in editing wikipedia at this stage at least, and if commenting on the talk page it is rather close to editing the encyclopedia. The reason being that she is so close to the material and it makes it rather hard to be objective or to know where to stop and how much to put into wikipedia or to make judgements about what is OR.

You must have some of the same issues yourself. I have that with some other topics in wikipedia, that it takes care because I am so close to it myself. But not so much here as I'm not at all involved in tuning theory development myself. I find the maths interesting, but it has moved way beyond any possibility of me contributing to the development of the maths itself unless I dedicated a fair bit of time to it, which I prefer to spend on other things like programming, writing my articles etc, given how many capable tuning theorists there are. And interested in the history in a general way also but that's not for me either to immerse myself in the historical part of the subject either.

So I feel I can be reasonably objective / neutral in wikipedia in this topic area, as much as anyone could be anyway. Robert Walker (talk) 01:22, 26 July 2016 (UTC)
 * Robert, a few rapid comments:
 * • I don't see why "meantone" should be reserved to "the entire gamut of regular diatonic tunings", which apparently are defined as tunings of seven notes in the octave, while we have been discussing 31-note meantones. Meantone, to me, is a name for any regular temperament.
 * • I must confess being unable to figure out what expressions as "1/4 undecimal comma meantone" might mean. (That is to say, I suppose I might make an effort to understand and succeed, but I won't.) I find such expressions counter-intuitive, I think that they have little historical justification and cannot concern historical meantone temperaments.
 * • Positive and negative systems are defined in the Glossary that opens Barbour's book on Tuning and Temperament as having fifths with ratios larger or smaller, respectively, than 3:2. He does not clearly say where these terms come from, but appears to suggest that they come from Bosanquet, when he writes (p. 125) that "Bosanquet gave a clear and comprehensive treatment of regular systems, both positive and negative, with a possible notation for them." In general I think that clarifying the origin of terms (especially when the references are available) makes them more understandable for the average reader.
 * • 700 cents might indeed be a better transition point than 3:2 because, as Margo says, it is the point at which the order of diatonic vs chromatic semitone (limma vs apotome) is inverted: positive systems have a chromatic semitone larger than the diatonic one, and negative system the inverse. But I have no reference for such a description.
 * • Did I write "al-Farabi"? This obviously is a slip of the pen and I thank you for correcting it. I meant (Safi al-Din) al-Urmawi, of course. His 17-note Pythagorean tuning is described also in several other Arabic manuscripts. I am not aware of a system described by him that would give a Wusta Zalzal at 72:59. Safi al-Din apparently thought that the 17-note division produced neutral intervals, say, that C-D♯ would habe been a neutral third; but his tuning was Pythagorean, and D♯ is only one comma higher than E♭, while the distance of a true neutral (Zalzalian) third would have been about two commas higher than the minor one. This however appears to be the origin of the schismatic tunings, not only those by Helmholtz and Ellis, but also some mentioned by Arnaut de Zwolle (and others) in the 15th century, as described by Mark Lindley.
 * • I am reluctant myself to getting involved in these WP articles: I let you do first, and may later add some more historical information, for instance about terminology, but I'd have to reread quite a few books (Bosanquet, in particular) and I have no time for that just now.
 * Hucbald.SaintAmand (talk) 07:35, 27 July 2016 (UTC)

Hucbald, okay a few remarks again.


 * Good cite on positive and negative temperament from Barbour Tuning and Temperament, glossary, page 15:


 * ""Negative System — A regular system whose fifth has a ratio smaller than 3:2. Positive System — A regular system whose fifth has a ratio larger than 3:2""


 * I agree, that though 700 cents is a more natural dividing line theoretically, I don't think we can use it in wikipedia unless we find a cite for it.


 * I'm familiar with the 17-note Pythagorean tuning. It's the next moment of symmetry (in Erv Wilson's terminology) after 12, as you continue the cycle of pure fifths - next time you get a system with only two step sizes. It's a subset of 53-et almost exactly, but it's not connected with 17-et in any way as the two step sizes are very uneven in size. I'm also interested to hear the details of the system they used from Margo.


 * On the range of meaning of meantones - if we take it in that general sense of what Margo calls an "eventone" rather than its historical sense - regular diatonic tunings would be by definition subsets of regular twelve tone tunings, because you could continue the cycle of fifths to obtain F C G D A E B F# C# G# D# A# and the interval F#-G is the same as B - C etc. So no matter how small the semitones get, you can still complete the construction to get a twelve tone system so long as their size is non zero. Does this answer you there? It might be worth making this point in the article, will look at it when I next work on editing it.


 * On the commas, I should have given the factorization, then it would be clearer. There's excellent online fast factorizer here


 * so "1/4-(896/891) meantone" - that's 2^7*7/(3^4*11) so after going C G D A E for the 3^4, and achieving the full comma at that point, then the result will be 3^4*7/(3^4*11) ignoring octave equivalence, the 3^4s cancel, giving you 7/11, or up to octave equivalence, 14/11. So the maths is simple really.


 * Similarly for "1/3-(352/351) meantone" factorized as 2^5*11/(3^3*13) then if you go up C G D A for the 3^3, you get to an interval of 13/11 for A to C.


 * For "1/14 (14680064/14348907) meantone" factorized as 2^21*7 / 3^15 then that means that after 14 fifths, achieving the full comma at that point again, you get a 7/3, or up to octave equivalence, 7/6.


 * I hope that's a bit clearer.


 * On the article, okay fine, I'll be doing a bit more editing of this article in a little while, in a few days time perhaps, based on our discussion here so far. I have a fair bit on as well and it's already much improved on what we had before. I think I'll remove the "under construction" template. As there's nothing that either of us can spot AFAIK that is misleading though there is plenty more than needs to be added to it. I'm happy with doing that for now, and for as long as it is only one article. The whole microtonal area is much in need of revision like this, while working on this I saw other articles that need work similarly. But I think that has to wait until we get more editors involved. I think there's a lot of work needed in this topic area, but fixing this article is a good start :).


 * I think one of the main remaining questions is about the meaning of positive and negative. I'm reluctant to use the one for 3/2s, even though that's our only cite, as it seems potentially confusing when the natural division is 700 cents for the topic of regular diatonic tunings, I think, based on Margo's explanation, so I'm inclined to just leave that for future research / discussion / clarification - and not do anything about it quite yet. I can just describe the situation I think without giving it a name. Will see how it goes. Robert Walker (talk) 12:32, 27 July 2016 (UTC)

Shorter summary for use on other pages
Since this page is summarized elsewhere in wikipedia using the idea of a syntonic temperament, we need a short summary we can use in its place. I've tried this out in the equal temperament page here to get us started: Equal temperament#Regular Diatonic Tunings.

I've also done a shorter version in the page on Musical Tunings here: Musical tuning and have updated the Template:Musical tuning.

Eventually we should change all the pages that link here so that they refer to these tunings as regular diatonic tunings instead of syntonic temperament as per the discussion above. I've done a minor edit along those lines here 31 equal temperament. It's easiest to find those pages with a google site search for "syntonic temperament" as a "what links here" check lists all the pages which use the Template:Musical tuning Robert Walker (talk) 02:06, 24 September 2016 (UTC)

Typo: "A chain of three fifths generates a minor third (A, D, G, C)"
"A chain of three fifths generates a minor third (A, D, G, C)" This is a chain of three fourths, isn't it? 79.79.171.35 (talk) 13:07, 3 January 2017 (UTC)


 * Yes you are right, thanks for the correction, fixed now. Robert Walker (talk) 14:59, 3 January 2017 (UTC)
 * Well, as a matter of fact, it does not really matter. We are talking here of pitch-classes, not actual pitches. I terms of pitch-classes, say, A-D is a fourth or a fifth; in terms of actual pitches, it is an ascending fourth or a descending fifth. But three ascending fourths produce a minor tenth (= octave + third), not a third, and three descending ones produce a major thirteenth (=octave + major sixth). Tricky, isn't it? Hucbald.SaintAmand (talk) 15:54, 3 January 2017 (UTC)
 * Good point. Just added " - in all those examples the result is reduced to the octave" Robert Walker (talk) 20:36, 3 January 2017 (UTC)
 * Assumption of ascending pitches, rather than pitch-classes, inferred from context: "'A chain of four equal sized fifths (E.g C, G, D, A, E) generates a major third consisting of two whole tones""A chain of three fourths generates a minor third (A, D, G, C)'"79.79.170.213 (talk) 13:24, 7 January 2017 (UTC)


 * == Plamondon's edit ==


 * Gentlepersons,


 * I apologize for not paying more (any!) attention to this article over the last few years. I trust that the edits that I made today answer the many excellent and cogent questions that y'all raised in the discussion above. I apologize for having the initial article be so unclear.


 * When we were first writing this content, it seemed haughty to insert it into any existing Wikipedia article, so we wrote it to be stand-alone (and made minor additions to other articles to refer to it). I am honored that you considered its content to be worthy of inclusion into Wikipedia's article on regular diatonic tunings.


 * I am no longer researching this topic. However, I encourage you to contact the lead researchers on this work, Dr. Bill Sethares (U. Wisconsin) and Dr. Andrew Milne (Western Sydney University), regarding any questions you might have.


 * Respectfully,


 * Jim Plamondon


 * P.S.: The key point is that the regular diatonic tunings aren't a set, they're a continuum, on which an infinite number of points fall. The tuning range of this continuum is narrow if one's timbre is harmonic (and one wants consonance, which, of course, one does). To widen the tuning range (all the way to the endpoints, which are culturally-important non-Western tunings), one need only align the current timbre's partials with the current tuning's notes (within the chosen temperament, which defines the mapping of notes to partials & vice versa). Do that, and you can have consonance along the temperament's entire tuning range. Or, you can choose to intentionally mis-align a note's partials, thereby introducing dissonance "on demand." Or, whatever: lots of new effects. JimPlamondon (talk) 13:16, 28 March 2017 (UTC)


 * Rightio. I don't understand though, why have you removed the text connecting it to regular diatonic tunings and linear temperaments? That also is a continuum. The syntonic temperament seems to equate to:


 * Regular diatonic tuning which have the tone and semitone of arbitrary size, including either of them zero, but always with the T T S T T T S pattern
 * Plus timbre mapping to make the more extreme regular diatonic tunings sound consonant.


 * The definition using fifths and octaves is equivalent to that.


 * Am I missing anything there? If that's the case, then the term "Regular diatonic tuning" is in much more common use than "Syntonic temperament" so it would seem to make sense to explain "Syntonic temperament" by reference to "Regular diatonic tunings" as this article did before your edit. The rest is probably fine as is, I can check the details to make sure but I think your section needs an introductory paragraph or sentence explaining the connection to Regular diatonic tuning.


 * This is also why I removed mentions of syntonic temperament elsewhere and replaced them with "Regular diatonic tuning" as the more common word. I can understand why you called it a syntonic temperament because the idea through timbral remapping is to make it sound like a tuning with the syntonic comma eliminated. But it is not a term in common use and it can be confusing at times because the term is intended for tunings that are meant for use with timbral remapping, but often when theorists and composers use various temperaments they are meant to be used "as is" without timbral remapping. So it is potentially confusing to classify them as "syntonic temperaments". Hope you understand why we did this. That's also why I did a new version of your diagram labeled as "Regular Diatonic Tunings" because it was being used throughout Wikipedia to illustrate articles that had nothing to do with timbral remapping.


 * I write this as someone who is myself interested in timbral remapping :). Robert Walker (talk) 13:55, 28 March 2017 (UTC)


 * Jim Plamondon writes: I apologize for deleting the text connecting the two definitions. That was inadvertent.


 * The definition of the syntonic temperament does not mention timbre, intentionally. The main value is that it facilitates a focus on the continuity of the string, rather than on the discrete beads. (By analogy, in physics, shifting focus from particle to field.) When focusing on the string, maintenance of consonance across the entire string is facilitated by mapping partials to notes...but this is a consequence of the change in focus, not a requirement of the definition of the syntonic temperament.


 * When introducing a new concept, as we were doing, one is always faced with a challenge: what word should be used to describe the concept? Use a neologism, or redefine an existing word? We decided that the strict definition of "temperament" as "a tuning system" (as opposed to a specific tuning) fit our intended usage quite well, especially since (in our usage) a temperament is defined by a comma sequence, hence tying everything together rather elegantly. Inevitably, this use of "temperament" is unfamiliar. So would be the definition of a neologism. It's a trade-off.JimPlamondon (talk) 06:37, 30 March 2017 (UTC)


 * Oh, okay, then I'll re-introduce the text connecting the two. A short paragraph at the head of your section should be sufficient I think. I think your distinction between a continuum and "beads" is more of a matter of emphasis. Tuning theorists always have thought of it as a continuum. They may talk about particular points like quarter comma or sixth comma, but always thought of linear temperaments as a continuum, as shown most strikingly by examples such as tuning using the golden ratio itself and the many phi tunings - those are not diatonic tunings of course, but there's also the linear tuning that achieves a golden ratio between the semitone and the tone in "golden meantone" and PI based Lucy Tuning. Perhaps it may help the article to mention examples like those somewhere in case the reader comes away with the impression that it is a discrete set of tunings rather than a continuum. I agree it is important to make clear that it is a continuum. Anyway Wikipedia goes by notability and established usage and so far your term "syntonic temperament" hasn't been taken up by the tuning community at large and the only cites I found were in the context of timbre remapping. While "Regular Diatonic Tuning" has much wider usage. The guideline is to go by the current situation rather than to try to predict the future (I'm not sure where, can look it up but it's somewhere in the guidelines).


 * And the thing is that normally syntonic comma tunings are thought of as occupying a particular range, in particular although you can use fractional and negative syntonic comma to express any tuning you like in the range of the regular diatonic tunings, it is very unusual to use this approach to describe schismatic temperaments or tunings with the fifth tuned slightly wide of pure. So, unless it becomes standard usage, which it hasn't yet, it's going to confuse the reader to refer to schismatic temperaments, for instance, as a special case of syntonic temperament even though mathematically they can. After all, mathematically syntonic temperaments and indeed all the tunings in the entire regular diatonic tuning could all be treated as special cases of schismatic temperaments too.


 * Hope that all makes sense. I'll add in that extra paragraph and look over what you've written and see if I have any more questions. Are you okay with what I just said? Robert Walker (talk) 09:02, 30 March 2017 (UTC)


 * Jim Plamondon writes: I appreciate your continued attention to this matter. :-)


 * I have a question: What metric did you use to determine that the term of art, "regular diatonic tuning," has a "much wider usage" than the term "syntonic temperament"? I wonder, because my preferred metric -- a Google search for the phrase in question -- returns 1,280 hits for "syntonic temperament" but only 695 for for "regular diatonic tuning". I am demonstrably not an experienced editor of Wikipedia articles, so I am eager to learn what the preferred metrics are, and their application to this example.


 * I agree wholeheartedly that one would not "use a fractional syntonic comma to describe schismatic temperaments or tunings," but I suspect that my reasons for not doing so are completely different from your reasons for not doing so. Both of these temperaments are generated by the tempered P5, so I would describe each specific tuning, uniquely, in terms of (a) its temperament (including its comma sequence), and (b) the width [in cents] of the tempered P5 (since that's the generator, the value of which changes smoothly across the temperament's valid tuning range).The first three sections of the following paper, referenced in the article above, explain this: http://sethares.engr.wisc.edu/paperspdf/tuningcontinua.pdf That article also discusses the "valid tuning range" of any temperament as we define it. This may clarify the issue of overlapping ranges among temperaments.


 * However, I am by far the junior author of these papers. Please allow me to bring my esteemed co-author, Dr. Andrew Milne, into this conversation. I suspect that his continuing attention to this matter, over the last decade (while my attention was elsewhere), will enable him to explain it far better than I.JimPlamondon (talk) 10:25, 30 March 2017 (UTC)


 * I will briefly just say a few things -- I haven't and time to read this full thread or review the articles changes. The term "syntonic temperament" -- to describe a range of diatonic tunings from 685.714 cents to 720 cents fifths -- has been used in Milne, A. J., Sethares, W. A., and Plamondon, J. (2007). Isomorphic controllers and Dynamic Tuning: Invariant fingering over a tuning continuum. Computer Music Journal, 31(4):15–32, and in Milne, A. J., Sethares, W. A., and Plamondon, J. (2008). Tuning continua and keyboard layouts. Journal of Mathematics and Music, 2(1):1–19. So above comments that this term is found only ina blog are incorrect. The 685-720 cent tuning range corresponds to what we denoted the 5-limit valid tuning range (VTR) in the second of those two papers. This is the tuning range over which all 5-limit consonances are correctly ordered by size (i.e., the tempered 3/2 is larger than the tempered 4/3 is larger than the tempered 5/4 is larger than the tempered 6/5). We specifically used the designation "syntonic" rather than meantone because the latter is usually used to refer to tunings from about 19-TET to 12-TET, while the syntonic temperament or continuum was specified as existing over the broader 685-720 cent 5-limit VTR. Perhaps also worth noting that this page shows the VTR for "meantone" as being 685-720 -- http://xenharmonic.wikispaces.com/Meantone+family. Having said that, the syntonic section Jim recently added needs a bit of tidying and some corrections (e.g., Wicki-Hayden is not "the best" layout for the syntonic continuum, but it is one of a few sensible choices). I'm off camping now for a few days... I'll aim to do some edits early next week. .../// Andy Milne ///...  — Preceding unsigned comment added by 122.106.107.68 (talk) 11:57, 30 March 2017 (UTC)


 * Jim and Andrew, the problem with google searches is that it can include anything including blog posts, many copies of wikipedia itself (there are many mirrors or wikipedia that are updated only infrequently), and so on. Wikipedia is supposed to be based on reliable sources. What counts as a reliable source depends on the topic matter. Also they might not even be about music. Many of the hits for "syntonic temperament" are about the term's use in psychology, I don't know what it means but it's not about music. E.g. "Normally this patient was of syntonic temperament, extroverted, "happy-go- lucky," with manifold interests and activities and a warm relationship to her husband and other members of her family".


 * Anyway, I think it's clear we are all talking about the same range of tunings here. So, it's not really a question about how to define them - except as a question about exposition. The question is what to call them. The article seems to give only one source for the phrase "Regular Diatonic Tuning", this thesis . I thought I had more than that but this is written a fair while back, I don't know if I had more sources and forgot to include them, I can look to see if I can find more, or ask friends if they know of more.


 * You give several sources but they are all by the same authors, so I don't think that's enough to establish that the term "syntonic temperament" has a wider usage than in your papers. Anecdotally microtonal friends, including composers, had no idea what "syntonic temperament" means and found the wikipedia articles confusing when they used this term. But of course that is not a basis on which to decide what word to use here.


 * I'll answer inline...


 * Yes, but they are proper peer reviewed journal articles!


 * The problem is that it is often a bit hard to find good or numerous sources for microtonal tuning theory as there is much less by way of scholarly articles than you'd expect from a topic that is under such active development - many composers for instance don't write scholarly papers.


 * I've just given you two examples.


 * We are not supposed to use Wikipedia to impose a new terminology on a field if it has not been widely accepted yet. And I don't think we can say that you have yet established "syntonic temperament" as the terminology for this range of tunings. But unless we find more sources, perhaps it is fair to say that "Regular diatonic tuning" hasn't been established in reliable sources either.


 * Indeed. But we have to call it something.


 * If neither has been established, and given that we have to use some word to describe it, then the range "Regular diatonic tuning" or "Syntonic temperament" whatever we call it, is much wider than for meantone surely, as you wouldn't use negative amounts of comma to construct a meantone scale with the fifth wider than pure.


 * So, then, using the term "syntonic temperament" to describe a tuning with a negative amount of the comma seems particularly confusing, and would make exposition difficult, as you'd have to keep explaining these points. Also "regular diatonic tuning" is a phrase that pretty much describes what it is, a tuning that has a regular pattern of tones and semitones as for the diatonic tuning. It also seems especially confusing if it means you have to say that a shizmatic temperament is a syntonic temperament, or indeed any tuning that is not constructed using the syntonic comma, but some other comma. Because it is most natural to describe the tuning using its own native comma, and why should they all also be defined with reference to the syntonic comma?


 * It shouldn't be confusing. The comma defines the mapping from JI to the tuning system. The VTR defines the tuning range over which the tempered consonances are correctly ordered by size (as detailed in Milne, A. J., Sethares, W. A., and Plamondon, J. (2008). Tuning continua and keyboard layouts. Journal of Mathematics and Music, 2(1):1–19 and reiterated here http://xenharmonic.wikispaces.com/Tuning+Ranges+of+Regular+Temperaments).


 * Of course if the term was in general usage to describe such tunings we'd have to use it, however confusing the exposition might be at times. But I don't think there is enough evidence to establish that.


 * And especially, the way it is used at the moment seems to be tied to timbral remapping, and that it requires us to call a tuning with a negative amount of the syntonic comma a "syntonic temperament" - all that makes sense in the context of timbral remapping, but, do you not think that it is potentially confusing outside of that context? Why would you want to use the amount of the syntonic comma to characterize a shizmatic tuning or one with the fifth wider than pure if you are not involved in timbral remapping? Anyway those are a few first thoughts. It's tricky, mainly because there is often so little published scholarly research on even the most basic of microtonal topics. Yet surely it is notable. Robert Walker (talk) 19:57, 30 March 2017 (UTC)


 * Temperaments are mappings from JI to lower rank tunings (which can be defined by a comma or list of commas). They do not have clear or precise boundaries over which they can be said to have perceptual validity -- probably dependent on person and context. The valid tuning range and the "nice" tuning range are at least two principled boundaries. The notion that a given tuning can be interpreted as being an exemplar of more than one temperament should not be a problem -- the VTRs of different temperaments overlap. After all, in an equal temperament, one can smoothly "modulate" between different temperaments by changing which intervals and scales are used. The original definitions of Syntonic temperament and VTR etc in the journal papers made no mention (as far as I recall) of timbral remapping, I don't know why that is even in the article. ...\\Andy Milne\\...


 * Andy, well the current version is substantially rewritten by Jim. But the original is here and it is based on this article:


 * The abstract for this paper says
 * ""This paper describes Dynamic Tonality, a system of realtime alterations to tuning and timbre that extends the framework of tonality to include new structural resources such as polyphonic tuning bends, tuning progressions, and temperament modulations. These new resources could prepare art music for the 21st Century""
 * ""This paper describes Dynamic Tonality, a system of realtime alterations to tuning and timbre that extends the framework of tonality to include new structural resources such as polyphonic tuning bends, tuning progressions, and temperament modulations. These new resources could prepare art music for the 21st Century""


 * This paper is what I was going on when I described it as being intimately connected with timbre remapping. It's true that there is no fixed boundary beyond which you stop using the syntonic comma to describe a tuning, but in one direction,there is a reasonably sharp boundary. It's my understanding that it would be rare to use it to describe a temperament with its tempered fifth sharper than the twelve equal fifth. A sharper fifth than that, flatter than 3/2 would be described using the schisma, as a Schismatic temperament. Calling a schismatic temperament a "syntonic temperament" seems particularly confusing for those used to it as being treated as a separate kind of temperament from meantones. And then tunings with the fifth shaper than 3/2 would not be referred to as schismatic or syntonic temperaments because they require negative amounts of the comma. Though it makes mathematical sense to use negative amounts of a comma, it just doesn't seem to be normal practice. I can't think of an instance of someone referring to a tuning using a negative fraction of a comma. Can you? Outside your own papers that is? It would just make the expositions here so complex, to have to explain that a schismatic temperament is also a syntonic temperament, and to introduce this historically novel idea of tempering by negative multiples of a comma, at least, as far as I know, it has no historical precedence. And I don't know of anyone outside of your papers that has used this idea of a syntonic temperament in this way.


 * Jim Plamondon writes: You appear to be confusing a temperament's valid tuning range with the temperament itself. Please allow me to quote from Milne 2007 (the most-often downloaded paper in the history of the Computer Music Journal, last time I checked):
 * A regular temperament... can be characterized by the small JI intervals called commas that are tempered to unison (Smith 2006). This means that a regular temperament is characterized by its temperament mapping, not its tuning, so any given temperament has a range of suitable tunings.
 * So, the fact that the valid tuning range of the syntonic temperament happens to include the valid tuning range of the schismatic temperament does not mean that we are, to use your words, "Calling a schismatic temperament a 'syntonic temperament'." We are not. The difference between temperaments is their comma sequence, not their tuning range. The fact that the syntonic temperament's tuning range includes the schismatic's temperament's tuning range is irrelevant (except that they both include P5=700, at which the syntonic temperament's major third is enharmonic with the schismatic temperament's diminished fourth, which makes that tuning a "pivot tuning" for temperament modulation between the syntonic and schismatic temperaments). The comma sequence can be used to inform the mapping of partials to notes of a tuning. The syntonic temperament, as per its comma sequence, maps the 2/1 partial to the octave, the 3/2 partial to the tempered perfect fifth, and the 5/4 partial to the major third, such that one would play a C-Ionian triad as C-E-G. The schismatic temperament, by contrast, maps the 5/4 partial to the diminished fourth, so that one would play the C-Ionian triad as C-F♭-G. Look at the Wicki–Hayden note layout. Do you see the G♭ button? The F♭ button would be just to the left of that, if the note-layout contained just two more notes per octave. (You can't assume that the D♯ and F♭ are enharmonic; that is only true for one particular point on the schismatic temperament's tuning continuum.) The Wicki-Hayden note-layout is very good for the syntonic temperament because it tightly groups the syntonic temperament's primary consonances; not so good for the schismatic temperament.--JimPlamondon (talk) 08:49, 29 April 2020 (UTC)


 * I can understand that you want to emphasize the continuous nature of these temperaments, especially with a background in timbre remapping. But I don't think, when it comes to a phrase to be used throughout wikipedia to talk about all tempered diatonic scales, that calling them all syntonic temperaments is the way to go, any more than calling them all schismatic temperaments. While calling them all Regular diatonic tunings is a way to both recognize the continuum that goes all the way from five equal to seven equal (not inclusive) while recognizing meantones, and schizmatic temperaments, as part of that continuum. Of course while referring to your own work then it makes sense to use your own term for these tunings. So my solution was to introduce the section describing your work with an explanation.


 * You can see how I did it in the introduction to the section here: Robert Walker (talk) 00:29, 31 March 2017 (UTC)


 * I suspect many of these issues can be resolved with a suitable careful rewrite, which I may attempt next week if I have time. I personally am not particularly bothered by the name of the page. The purpose of my last two responses was to correct some previous assertions that have cropped up in this discussion page. Just to quickly add that there is, arguably, a natural correspondence between the syntonic temperament and the 7 to 5 range because tempering out the syntonic comma is the only temperament (as defined by the commas it tempers out) that has precisely this VTR. And if this page is actually about diatonic (maximally even 5 large, 2 small) then syntonic temperament, schismatic temperament etc, are just overlapping ranges within this continuum. But, it's not at all clear what this page is even about, or how it relates to other pages. — Preceding unsigned comment added by 122.106.107.68 (talk) 02:54, 31 March 2017 (UTC)

Oh, just re-read what you said about "Valid Tuning Range", I get what you are saying there. Though the valid tuning range is a different thing from the range over which the syntonic comma is customarily used to define tunings. As after all it requires use of negative amounts of the comma apart from anything else. As for the scope of this page I lead out with a definition that the reader would find easy to understand which also makes it easy to understand why it is called a "regular diatonic tuning":

""any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's.""

That's a well defined continuum of tunings, and I'm not sure therefore how one can be unsure what it is about? I then go on to give the more technical definition from the thesis in which "Regular Diatonic Tuning" is defined, as

""Regular" here is understood in the sense of a mapping from Pythagorean diatonic such that all the interval relationships are preserved"."

They could be defined in many other ways too. But it's the same continuum of tunings, no matter how you define them. Just different definitions which mark out the same entities. I'm trained as a mathematician, so it might be that I am coming to this from a slightly different perspective, so that things that seem identical to me look different to others? For me the continuum of tunings is what this page is about, and how you define them doesn't really matter so long as you mark out the same tunings, and it just seemed that in an article addressed to the ordinary non mathematical reader, it's best to start off with the simplest easiest to understand definition. And of all the definitions, the TTS TTTS definition seemed by far the easiest to both state and to understand. That's the motivation for doing it that way. Robert Walker (talk) 03:56, 31 March 2017 (UTC)

Added sentence to lede explaining that any scale of form TTSTTTS is a linear temperament
I just added this which may help with exposition and to tie the article together:

""In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size (TTTS or a permutation of that) which makes it a Linear temperament with the tempered fifth as a generator.""

It's one of those things that seems self evident, to anyone with the musical background to understand the construction of a diatonic scale, like "2 + 2 = 4" with just a bit of reflection so haven't bothered with either citing it or giving a formal proof. Robert Walker (talk) 01:02, 2 April 2017 (UTC)

Proposal to restore Syntonic temperament as a stand-alone article
Gentlepersons,

Let me again thank you all for your learned attention to the matter of the "syntonic temperament." Again, let me emphasize that the lead authors of the seminal papers -- Prof. William Sethares and Dr. Andrew Milne -- both know infinitely more than I do about both music and mathematics. I am a mere marketing slime, who happened to read about the Wicki-Hayden keyboard and realized that its having "the same fingering in all keys" HAD TO BE the result of some correspondence between the geometric pattern of the notes and the deep structure of music. Reading Sethares' book "Tuning, Timbre, Spectrum, Scale" introduced me to the notion that consonance arose solely from the alignment of a tuning's notes with a timbre's partials. Everything else arose from those two observations, with nearly all of the heavy lifting having been done by Milne and Sethares (et al.) while I kibbitzed. So, whenever either of them corrects me, they are right, and I am wrong.

That said, please allow me to propose that Wikipedia's stand-alone article on the syntonic temperament should be restored, with only a brief summary of it included in Wikipedia's article on Regular diatonic tunings.

Here's why: The inclusion of the syntonic temperament in the "regular diatonic tunings" article is problematic, because it assumes that all tunings of the syntonic temperament are regular and diatonic, when, in fact, both of these assumptions are false.

First, "diatonic:" The valid tuning range of the syntonic temperament includes an endpoint at which the minor second has a width of zero (5-tet) and another endpoint at which it has a width equal to the major second (7-tet), neither of which are diatonic.

Second, "regular:" Consider the discussion of "Related Just Intonations" on the pages numbered as 79-80 in Spectral tools for Dynamic Tonality and audio morphing, which describes the Tone Diamond used in Dynamic Tonality synthesizers. Moving the control-point within the Tone Diamond systematically shifts the tuning away from regularity. To quote a paragraph on the page labeled as 80:
 * When the Tone Diamond’s control point is anywhere along the central horizontal line (the “Max. Regularity” line), the tuning is a one- or two-dimensional tuning such as 12-TET or quarter-comma meantone, as shown on the main tuning slider. When the control point is moved upward or downward the tuning moves towards a related just intonation. The tunings that are intermediate between perfect regularity and JI are like the circulating temperaments of Kirnberger and Vallotti in that every key has a (slightly) different tuning. And all of these tunings have the same fingering when played on a 2D lattice controller.
 * When the Tone Diamond’s control point is anywhere along the central horizontal line (the “Max. Regularity” line), the tuning is a one- or two-dimensional tuning such as 12-TET or quarter-comma meantone, as shown on the main tuning slider. When the control point is moved upward or downward the tuning moves towards a related just intonation. The tunings that are intermediate between perfect regularity and JI are like the circulating temperaments of Kirnberger and Vallotti in that every key has a (slightly) different tuning. And all of these tunings have the same fingering when played on a 2D lattice controller.

The syntonic temperament is therefore not, strictly speaking, a regular tuning, nor a family of regular tunings, nor even a regular tuning system. It embraces closely-related irregular tunings, too.

In summary: because the syntonic temperament embraces tunings that are not regular and not diatonic, having Wikipedia's only discussion of the syntonic temperament be subsumed into Wikipedia's article on "Regular diatonic tunings" is illogical and inappropriate. It attempts to squeeze a hypercubic discussion into a cubic hole.

That's why I initially put my description of the Syntonic temperament into its own Wikipedia article, and then added references from other articles to it (and vice versa), rather than attempting to shoehorn it into any existing article. This "where to put this discussion?" problem is just one small piece of evidence that the ideas embodied Dynamic Tonality, and hence in Musica Facta (the research project into Dynamic Tonality), the syntonic temperament, and Guido 2.0 (the aspect of Musica Facta focused on music education) may be incommensurable with those of music-making's current paradigm.

I apologize for not presenting these arguments when the idea first arose of merging the discussion of the syntonic temperament article into some other pre-existing article.

I hereby propose reverting the article on the Syntonic temperament to being stand-alone, and adding a brief summary of it, with a reference to the full article, in the Regular diatonic tunings article.

Respectfully,

Jim Plamondon --JimPlamondon (talk) 06:44, 26 April 2020 (UTC)

Relationship between tuning, timbre, etc.
Gentlepersons,

Thank you again for your continued attention to this matter. :-)

In the Talk above, people have repeatedly questioned the relationship between the syntonic temperament and timbre. Temperament, they say, has nothing to do with timbre; the linking of the two, which runs throughout Sethares, Milne, Plamondon, Prechtl, Tiedje, et al., they see as aberrant.

I think I now understand where y'all are coming from. I suspect that the microtonal community has assumed the use of Just Intonation and Harmonic Timbres for so long that they have forgotten that they are even making an assumption. They often write about ratios of small whole numbers, as Pythagoras did, as if the ratios themselves were the source of tonal meaning -- when it has been abundantly clear since Joseph Sauveur, or, for the doubtful, since Helmholtz, and inescapably since Plomp & Levelt that these intervals arise from the specific pattern of vibrations defined by the Harmonic Series, not from some magic of numerology.

Alternatively put: Sethares, Milne, Plamondon, et al. are explicitly relating tempered pseudo-Just tunings and tempered pseudo-Harmonic timbres, whereas microtonalists and previous music theoreticians have implicitly related [un-]tempered Just tunings and untempered Harmonic timbres.

Examples of the above implicit assumptions in the Talk section of this article:
 * Hucbald: "The possibility of negative temperament (fifths wider than pure) is a bit farfetched." Why are pure intervals being assumed? Because (I suspect) harmonic timbres are being assumed.
 * Walker: "But the chart shown includes tunings with the fifth wider than the pure fifth. So it's hard to be sure what it is about." Why are tunings wider than the pure (i.e., Just) fifth confusing? Because (I suspect) harmonic timbres are assumed.
 * Walker: "Hucbald, Okay, I see what you are saying, that any regular diatonic scale is a meantone in that more general sense that the tempered major third is equally divided into two equal tones. However, though that is logical, it doesn't correspond to established usage. As an example, 17 equal, with a fifth of 705.884 cents, so tempered by (705.884-701.955) or 3.929 cents wider than just, is -1/5.47 meantone but I don't think anyone calls it that, and it would also be almost exactly -2 schisma but again that doesn't sound right either." Walker is confusing tuning (that is, a specific width of the P5) with temperament-as-defined-by-a-comma-sequence (which determines the association between notes in a tuning and partials in a timbre). This "doesn't correspond to established usage" iff one ignores the many peer-reviewed publications of Sethares, Milne, Plamondon, Prechtl, Tiedje, et al., over the last 15 years. Please let me elaborate. It is true that, at P5=705.884, an untempered Just diminished fourth aligns excellently with the 5th partial of an untempered Harmonic timbre, and the association of the diminished fourth with the 5th partial is exactly what you'd get from a comma sequence (1/2, 5/4, schisma) -- that is, from theschismatic temperament-as-defined-by-a-comma-sequence. However, P5=705.884 is just a tuning -- that is, a specific width of the generator -- which could be within the valid tuning range of any of a large number of temperaments, each defined by a different comma sequence. I suspect that Walker's implicit assumptions are leading him to make a very specific and systematic error, as described above.


 * Hucbald: "I utterly fail to see in what sense something called 'timbre' could 'conform to the harmonic series'." This is an utterly pure example of an assumption that is so implicit that it is not even recognized as an assumption. Hucbald should, perhaps, read Sethares' "Tuning, Timbre, Spectrum, Scale," published in 1998, and now in its second edition. The content of this book makes the relationship between the terms in its title crystal clear. That the content of the Syntonic temperament article does not clarify the reader's implicit -- and, in this case, mistaken -- assumptions, is a problem for which I suggest a solution below.
 * Walker: " Anyway he [Gene Ward Smith] developed various tools that let composers see interconnections between tuning systems and to devise new tuning systems and to see individual tunings as part of larger patterns that they hadn't been able to see before. This has been a vibrant synthesis of maths and music. Much of it developed in just the last ten years. I'm a mathematician by training myself but it's one of the things about maths that it is very common that you don't understand the maths developed even by people who are working in very closely related areas. It is just the nature of it because of the way it works with such abstract ideas. But that doesn't prevent it from being valuable." We agree with this assessment of Gene Ward Smith's work, and indeed our definition of temperament-as-defined-by-a-comma-sequence is his invention. We would welcome y'all's affording the same courtesy to our our mathematics, documented in peer-reviewed journals such as the Computer Music Journal and the Journal of Music and Mathematics, which clearly (well, using mathematics, so it's only clear to those who speak the lingo) makes and proves, using mathematical tools, the validity and usefulness of our approach.
 * Hucbald: "We would easily agree, I trust, about what a temperament is: in addition to make use of tempered intervals, it is also meant for instruments of fixed tones, producing a limited number of tones." I do not, in fact, agree with this definition, and for very good reasons, as documented in our published papers. The fact that y'all are arguing about the meanings of these terms among yourselves, shows that there is ample ambiguity in them, into which we can offer definitions that are both narrower and more general, as we have done in our publications and in the original Syntonic temperament article. It is entirely possible that we have failed to be sufficiently clear in those papers, or that we failed to translate that clarity to our original stand-alone Wikipedia article. However, as Oliver Cromwell famously said, "I beseech you, in the bowels of Christ, think it possible that you may be mistaken." Particularly, I beseech you to take note of (what I suspect is) the microtonal community's -- and nearly all previous Western music theorists' -- assumption that Just Intonation tunings would be used almost exclusively with timbres that followed the Harmonic Series, and the impact of that assumption on your attempts to shoehorn any discussion of the Syntonic Temperament into a space that doesn't have enough dimensions to contain it.

Sethares, Milne, Plamondon, Prechtl, Tiedje, et al. have offered something that is truly novel, and y'all seem to be working very hard to deprecate its novelty and force it into the lower-dimensional framework of what has historically been done. Our work COMBINES:
 * The use of a temperament-as-defined-by-a-comma-sequence to define the mappings between the notes of a pseudo-Just tuning and a pseudo-Harmonic timbre (Example: In the syntonic temperament, the 5th partial is mapped to the major third; in the schismatic temperament, it is mapped to the diminished fourth);
 * Dynamic tuning within the valid tuning range of a pseudo-Just temperament-as-defined-by-a-comma-sequence, including tunings that are regular and irregular, equal and unequal, major-aligned and minor-aligned, which is in itself a significant advance in the state of the art;
 * Dynamic changes to timbre to align said timbre's pseudo-Harmonic partials with any said tuning of the notes of said temperament, and to systematically emphasize and/or de-emphasize sets of prime partials, and to systematically mis-align said partials away from said note, all of which are novel real-time musical effects;
 * Two-dimensional note-layouts (notably the Wick-Hayden note-layout) which enables consistent fingering within the valid tuning range of a rank-2 temperament-as-defined-by-a-comma-sequence, thus enabling a musician to perform novel musical effects such as those listed above, and also polpyphonic tuning bends resulting from the narrowing or widening of the generator in real time, which in turn enables novel tonal effects such as temperament modulations (for example, from syntonic to schismatic using P5-700 as a pivot tuning, because at P5=700, the diminished fourth and major third are enharmonic, enabling the temperament to be switched from the syntonic temperament's mapping of the 5th partial to the major third, to the schismatic temperament's mapping of the 5th partial to the diminished fourth, without a jarring change in timbre);
 * A means of changing the width of the generator, within the current temperament's valid tuning range, in real time.

All of these innovations have been documented in the peer-reviewed literature -- and some non-peer-reviewed technical reports -- for over a decade. Our Computer Music Journal article was still, the last time I checked, the most-often downloaded article in its history. Our "sight reading music theory" article has been downloaded from ResearchGate over 3500 times. To claim that these innovations, or the ideas that they document, or the terms that they use, are not "established," begs the question of what definition of "established" you are using.

When combined, our system's components enable a systematic expansion of the frontiers of tonality and novel tonal effects. However, each component by itself is of little particular interest. How useful is an automotive transmission without an engine? A computer without electricity? A browser without the Internet? So, our explanation of each component explicitly refers to the other components.

Y'all, I suspect, are not making such explicit connections among your uses of "temperament," "timbre," etc., because youimplicitly assume:
 * That the goal of a tuning is to maximize the alignment of its notes' fundamentals with the ratios of Just Intonation;
 * That such tunings will be used with timbres that conform to the Harmonic Series; and
 * That these tunings and timbres will be played on the piano keyboard and/or instruments dedicated to a particular fixed tuning.

In summary, we are explicitly stating relationships that you are assuming implicitly.

The solution is to update all of the other articles on microtonality and music theory to make their implicit assumptions explicit.

Y'all are resisting this. Your resistance is manifesting itself in an attempt to shoehorn our innovations into a box defined by your lower-dimensional implicit assumptions, in which the interconnections and synergies must be stripped away to fit in those lower-dimension boxes. This looks to me to be a clear-cut case of the Semmelweis Reflex. That reflex is a natural human and institutional reaction to novel ideas, so I don't blame y'all in any way for exhibiting it. I have every confidence that I am displaying similar human reflexes and cognitive biases. :-)

I am happy to see that the discussion surrounding the original Syntonic temperament article led to the creation of the Regular diatonic tunings article, which is a boon to Wikipedia.

As a first step towards resolving the "problem of assumptions" described above (and the fact that the Syntonic temperament includes non-regular and non-diatonic tunings), I will de-merge the Syntonic temperament article out of the Regular diatonic tunings article sometime in early June 2020.

In the meantime, I will add to the article on Dynamic Tonality a section that discusses (a) the implicitly-assumed relationships described above, and (b) the explicitly-defined relationships made by Dynamic Tonality. I can them update the discussion of the Syntonic Temperament to refer to this section in the Dynamic Tonality article. I can then add a reference to this same "implicit assumptions" section to all other articles on traditional music theory. This will clarify the distinction between Dynamic Tonality and everything that came before.

That should resolve all of the issues raised in this Talk page, and improve Wikipedia's discussion of music theory considerably.

I look forward to your comments on this suggestion.

Respectfully, --JimPlamondon (talk) 04:49, 2 May 2020 (UTC)

Either remove article or remove everything related to Syntonic Temperament and Dynamic tonality
Maybe about 80% of this article seems to correspond to a fringe theory by user JimPlamondon. This is against wikipedia's philosophy, as you can see here. In the fringe theories page. This has nothing to do with whether I agree or disagree with this theory, it is an objective fact that this is not broadly supported and as such should be deleted. Wikipedia is not a place for promoting your own theories, it should reflect what is broadly acceped in a field, there are many other websites that can be used to promote or support this research.

In a nutshell, as my reference states: To maintain a neutral point of view, an idea that is not broadly supported by scholarship in its field must not be given undue weight. I don't think this deserves extra discussion, and JimPlamondon's own comments on the matter make this very clear. I will procede to delete everything associated with Dynamic Tonality IgnacioPickering (talk) 23:52, 17 July 2021 (UTC)

Ignacio, et al.: Wikipedia states that "fringe theories in science depart significantly from mainstream science and have little or no scientific support." The content that you deleted is, by that definition, not a fringe theory, in that it is well-supported by numerous publications in the peer-reviewed scientific literature. Indeed, the content that you deleted contains more references (7) that the content that you did not delete (4).

This leads to the second criteria for identifying fringe theories: Parity of sources. Four of the seven sources cited in the content you deleted are to respected peer-reviewed publications, such as the Computer Music Journal (H-Index: 41), the Journal of Mathematics and Music (H-Index: 14), and the Proceedings of the College Music Society. Only three of the seven sources are not similarly peer-reviewed. By comparison, of the four sources cited in the non-deleted content, the first refers to a master's thesis; the second to a non-peer-reviewed book; the third to a non-peer-reviewed review of said book; the fourth to a non-peer-reviewed wiki post. None but the first of these four sources can credibly be claimed to be peer-reviewed. The other three can reasonably be categorized as "self-published texts," which Wikipedia's description of fringe theories cites as a sign of fringe theories. In short: The deleted content is demonstrably less fringy than the non-deleted content.

Please note also that publications on the syntonic temperament and Dynamic Tonality are cited frequently, according Google Scholar:

This is an impressive level of citation for papers in the humanities (such as music theory), in which [https://www.nature.com/articles/d41586-017-08404-0#:~:text=The%20Web%20of%20Science%20records%2C%20for%20instance%2C%20that%2065%25%20of%20humanities%20papers%20published%20in%202006%20have%20not%20yet%20been%20cited. 65%] of papers are never cited. These citations prove the "broad acceptance" of the content that you deleted.

I would go a step further, and point out that many of the papers supporting the syntonic temperament and Dynamic Tonality include mathematical proofs of their claims. Such proofs are the gold standard for claims made about models of phenomena in the natural world, such as musical tunings. The citations supporting the non-deleted content contain no such mathematical proofs. Therefore, the theory of the non-deleted content is less strongly supported by its citations than is the content that you deleted.

Finally, as Wikipedia's discussion of fringe theories states, "One important barometer for determining the notability and level of acceptance of fringe ideas related to science, history or other academic pursuits is the presence or absence of peer-reviewed research on the subject." As shown above, the content that you deleted meets this standard much better than the content that you did not delete.

Therefore, the deleted content should have been retained, and the non-deleted content should have been deleted as a fringe theory.

However, Ignacio, I am not unreasonable. I will not press for the deletion of the obviously-fringy theory represented by the content that you did not delete. I will only insist that you follow Wikipedia's rules fairly, and therefore not delete the material on the syntonic temperament and Dynamic Tonality. --JimPlamondon (talk) 03:27, 18 July 2021 (UTC)

Incorrect description of syntonic comma
The article claims: "The small difference in pitch between [a chromatic and diatonic semitone] is called a comma, usually prefixed by the name of the tuning system that generates it, such as a syntonic comma (21.5 ¢)..."

This is clearly incorrect. While the Pythagorean comma does work out such a difference, as seen on the corresponding page, the syntonic comma is not the difference between the chromatic (25:24) and diatonic (16:15) semitones of just intonation, equivalently the minor second and augmented unison - that yields 128:125, i.e. the (enharmonic) diesis.

The syntonic comma (81:80) - the one referred to when one speaks of "fractional-comma meantone" - is a ratio between the major tone (9:8) and minor tone (10:9) of just intonation - which is not a regular diatonic tuning, as it uses two different tones. Equivalently, it's the difference between a just-intonation ditone and major third.

216.209.47.135 (talk) 07:57, 31 January 2024 (UTC)